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                                                                    Exhibit 99.2
 
                                  NEWS RELEASE
 
FOR IMMEDIATE RELEASE:                                   CONTACT: SUSAN PIROTINA
                                                                  (210) 283-2631
 
                       TESORO ANNOUNCES PRIVATE OFFERING
                          OF SENIOR SUBORDINATED NOTES
 
     SAN ANTONIO -- JUNE 1, 1998 -- Tesoro Petroleum Corporation (NYSE: TSO)
announced today that it plans a private offering of $300 million in Senior
Subordinated Notes eligible for Rule 144A in connection with its plans to
finance its recent refinery acquisitions.
 
     The Company recently completed the acquisition of a refinery and 32 retail
marketing outlets in Hawaii from The Broken Hill Proprietary Company Limited and
has agreed to acquire a refinery in Anacortes, Washington from Shell Refining
Holding Company. In connection with these acquisitions, the Company intends to
raise proceeds from certain previously announced public equity offerings and the
proposed offering of Senior Subordinated Notes.
 
     The Senior Subordinated Notes are expected to have a ten-year maturity
without sinking fund requirements. The Notes will be subject to optional
redemption by the Company after five years at declining premiums.
 
     The Senior Subordinated Notes will not be registered under the Securities
Act of 1933, as amended, and may not be offered or sold in the United States
absent registration or an applicable exemption from registration requirements.
 
     Tesoro Petroleum Corporation is a natural resource company engaged in
petroleum refining and marketing, natural gas exploration and production,
marketing and distributing petroleum products and providing marine logistics
services.
 
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