UNITED STATES                      
                      SECURITIES AND EXCHANGE COMMISSION         
                           WASHINGTON, D.C. 20549                 
                                              
                                              

                                 SCHEDULE 13G


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

                           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
- ---------------------                                   ------------------------
 
- --------------------------------------------------------------------------------
1      NAME OF REPORTING PERSON
       S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON
 
       Dodge & Cox                            94-1441976
 
- --------------------------------------------------------------------------------
 2     CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP*         (a) [_]
                                                                 (b) [_]

       N/A

- --------------------------------------------------------------------------------
3     SEC USE ONLY



- --------------------------------------------------------------------------------
4     CITIZENSHIP OR PLACE OF ORGANIZATION

      California - U.S.A.

- --------------------------------------------------------------------------------
                 5     SOLE VOTING POWER

   NUMBER OF           4,510,926

    SHARES       -----------------------------------------------------------
                 6     SHARED VOTING POWER
  BENEFICIALLY
                       55,700
   OWNED BY
                 -----------------------------------------------------------
    EACH          7    SOLE DISPOSITIVE POWER

  REPORTING            4,971,116

    PERSON       -----------------------------------------------------------
                  8    SHARED DISPOSITIVE POWER
    WITH
                       -0-

- --------------------------------------------------------------------------------
9     AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON

      4,971,116

- --------------------------------------------------------------------------------
10    CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN SHARES*

      N/A

- --------------------------------------------------------------------------------
11    PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9

      5.4%

- --------------------------------------------------------------------------------
12    TYPE OF REPORTING PERSON*

      IA

- --------------------------------------------------------------------------------
                     *SEE INSTRUCTION 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., 35th 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:
                         ------------------------- 
                         4,971,116

                    (b)  Percent of Class:
                         ----------------       
                         5.4%


                               PAGE 3 OF 4 PAGES

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

                    (i)    sole power to vote or direct the vote: 4,510,926

                    (ii)   shared power to vote or direct the vote: 55,700

                    (iii)  sole power to dispose or to direct the disposition 
                           of: 4,971,116

                    (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 12, 1998
       -----------------
                          DODGE & COX

                          By:     /S/ W. TIMOTHY RYAN
                             ---------------------------------------
                          Name:   W. Timothy Ryan
                               -------------------------------------
                          Title:  Secretary-Treasurer
                                ------------------------------------

                               PAGE 4 OF 4 PAGES