------------------------------   
                                             OMB APPROVAL
                  UNITED STATES              OMB NUMBER 3235-0145
        SECURITIES AND EXCHANGE COMMISSION   EXPIRES:  DECEMBER 31, 1999
            WASHINGTON, D.C. 20549           ESTIMATED AVERAGE BURDEN
                                             HOURS PER RESPONSE . . . 14.90
                                             ------------------------------

                                 SCHEDULE 13G


                   UNDER THE SECURITIES EXCHANGE ACT OF 1934
                            (AMENDMENT NO. __1__)*

                           Amerada Hess Corporation
- - ------------------------------------------------------------------------------
                               (Name of Issuer)

                                    Common
- - ------------------------------------------------------------------------------
                        (Title of Class of Securities)

                                   023551104
                  -------------------------------------------
                                (CUSIP Number)


Check the following box if a fee is being paid with this statement [ ]. (A fee
is not required only if the filing person: (1) has a previous statement on file
reporting beneficial ownership of more than five percent of the class of
securities described in Item 1; and (2) has filed no amendment subsequent
thereto reporting beneficial ownership of five percent or less of such class.)
(See Rule 13d-7).

*The remainder of this cover page shall be filled out for a reporting person's
initial filing on this form with respect to the subject class of securities, and
for any subsequent amendment containing information which would alter the
disclosures provided in a prior cover page.

The information required in the remainder of this cover page shall not be deemed
to be "filed" for the purpose of Section 18 of the Securities Exchange Act of
1934 ("Act") or otherwise subject to the liabilities of that section of the Act
but shall be subject to all other provisions of the Act (however, see the
Notes).

                               PAGE 1 OF 4 PAGES

 
- -----------------------                                  ---------------------
  CUSIP NO. 023551104               13G                    PAGE 2 OF 4 PAGES
- -----------------------                                  ---------------------
 
- ------------------------------------------------------------------------------
      NAME OF REPORTING PERSON
 1    S.S. or I.R.S. IDENTIFICATION NO. OF ABOVE PERSON
                                                                                
      Dodge & Cox                  94-1441976                                  
                          
- ------------------------------------------------------------------------------
      CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP*
 2                                                              (a) [_]
                                                                (b) [_]
      N/A                                        
- ------------------------------------------------------------------------------
      SEC USE ONLY
 3
 
    
- ------------------------------------------------------------------------------
      CITIZENSHIP OR PLACE OF ORGANIZATION
 4                                 

      California - U.S.A.
- ------------------------------------------------------------------------------
                          SOLE VOTING POWER
                     5                                                       
     NUMBER OF            5,451,976
 
      SHARES       -----------------------------------------------------------
                          SHARED VOTING POWER
   BENEFICIALLY      6        
                          64,200
     OWNED BY
                   -----------------------------------------------------------
       EACH               SOLE DISPOSITIVE POWER
                     7      
    REPORTING             5,974,316
 
      PERSON       -----------------------------------------------------------
                          SHARED DISPOSITIVE POWER

       WITH          8       
                          -0-
- ------------------------------------------------------------------------------
     AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
 9                          

     5,974,316
- ------------------------------------------------------------------------------
     CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES*
10         
                                                                    [_]
     N/A
- ------------------------------------------------------------------------------
     PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)
             
11                                                                          
     6.6%
- ------------------------------------------------------------------------------
     TYPE OF REPORTING PERSON*
12       

     IA
- ------------------------------------------------------------------------------

                     *SEE INSTRUCTIONS BEFORE FILLING OUT!

                               PAGE 2 OF 4 PAGES

 
   Item 1(a)   Name of Issuer:
               -------------- 
               Amerada Hess Corporation

   Item 1(b)   Address of Issuer's Principal Executive Offices:
               ----------------------------------------------- 
               1185 Avenue of the Americas
               New York, NY 10036

   Item 2(a)   Name of Person Filing:
               --------------------- 

               Dodge & Cox

   Item 2(b)   Address of the Principal Office or, if none, Residence:
               ------------------------------------------------------- 
               One Sansome St., 35/th/ Floor
               San Francisco, CA  94104

   Item 2(c)   Citizenship:
               ----------- 
               California - U.S.A.


   Item 2(d)   Title of Class of Securities:
               ---------------------------- 
               Common

   Item 2(e)   CUSIP Number:
               ------------ 

               023551104

   Item 3      If the Statement is being filed pursuant to Rule 13d-1(b),
               ----------------------------------------------------------
               or 13d-2(b), check whether the person filing is a:
               ------------------------------------------------- 

               (e) [X]  Investment Advisor registered under section 203 of
                        the Investment Advisors Act of 1940

   Item 4      Ownership:
               --------- 
               (a)  Amount Beneficially Owned:
                    ------------------------- 
                    5,974,316

               (b)  Percent of Class:
                    ----------------       
                    6.6%

                               PAGE 3 OF 4 PAGES

 
            (c) Number of shares as to which such person has:
                ---------------------------------------------

            (i) sole power to vote or direct the vote: 5,451,976
 
           (ii) shared power to vote or direct the vote: 64,200
 
          (iii) sole power to dispose or to direct the
                disposition of: 5,974,316

           (iv) shared power to dispose or to direct the
                disposition of: -0-


 Item 5   Ownership of Five Percent or Less of a Class:
          -------------------------------------------- 
          Not applicable.

 Item 6   Ownership of More than Five Percent on Behalf of Another
          --------------------------------------------------------
          Person:
          ------
          Securities reported on this Schedule 13G are beneficially owned by
          clients of Dodge & Cox, which clients may include investment companies
          registered under the Investment Company Act and/or employee benefit
          plans, pension funds, endowment funds or other institutional clients.

 Item 7   Identification and Classification of the Subsidiary Which
          --------------------------------------------------------- 
          Acquired the Security Being Reported on By the Parent
          -----------------------------------------------------
          Holding Company:
          ---------------  
          Not applicable.

 Item 8   Identification and Classification of Members of the Group:
          --------------------------------------------------------- 
          Not applicable.

 Item 9   Notice of Dissolution of a Group:
          -------------------------------- 
          Not applicable.

 Item 10  Certification:
          ------------- 
          By signing below I certify that, to the best
          of my knowledge and belief, the securities referred to
          above were acquired in the ordinary course of business and
          were not acquired for the purpose of and do not have the
          effect of changing or influencing the control of the issuer
          of such securities and were not acquired in connection with
          or as a participant in any transaction having such purpose
          or effect.


                                   SIGNATURE

 After reasonable inquiry and to the best of my knowledge and belief, I certify
that the information set forth in this statement is true, complete and correct.

Dated: February 10, 1999
       -----------------

                                   DODGE & COX

                                   By: /s/ W. TIMOTHY RYAN
                                     ---------------------------------------
                                   Name: W. Timothy Ryan
                                        ------------------------------------
                                   Title: Senior Vice President
                                         -----------------------------------

                               PAGE 4 OF 4 PAGES