The 25th Amendment
If the President of the United States resigns or dies in office and the Vice-President becomes President, what is the procedure for selecting a new Vice President? If I remember correctly, that's what happened when Spiro Agnew resigned and Ford became the new Vice President under Nixon.
The 25th Amendment to the US Constitution (ratified February 10, 1967) specifies the following procedure:
Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress.
Ford, who at the time of his nomination was serving in the U.S. House of Representatives, was the first Vice President to benefit from the Amendment, which he had helped to draft. Nominated by Nixon on October 12, 1973, he was confirmed by Congress on December 6 of that year. When Nixon resigned less than a year later (on August 9, 1974), Ford was immediately sworn in as the United States' 38th president - the first man to enter the White House without ever having won a national election.
Ford nominated Nelson A. Rockefeller as his vice president on August 20, 1974; Rockefeller was confirmed by Congress on December 19 of that year.
Check out the Information Please almanac for a complete list of the order of presidential succession, for the event that the presidency and vice-presidency are vacant at the same time.