Twenty-second Amendment to the United States Constitution
Some argue that the 22nd Amendment and 12th Amendment bar any two-term president from later serving as vice president as well as from succeeding to the presidency from any point in the presidential line of succession. Others contend that the original intent of the 12th Amendment concerns qualification for "service" (age, residence, and citizenship), while the 22nd Amendment, concerns qualifications for "election", and thus (strictly applying the text) a former two-term president is still eligible to "serve" as vice president (neither amendment restricts the number of times an individual can be elected to the vice presidency), and then succeed to the presidency to serve out the balance of the term (though prohibited from running for election to an additional term).
Vice President of the United States
The vice president is indirectly elected together with the president to a four-year term of office by the people of the United States through the Electoral College. Section 2 of the Twenty-fifth Amendment, ratified in 1967, created a mechanism for intra-term vice presidential succession, establishing that vice presidential vacancies will be filled by the president and confirmed by both houses of Congress. Previously, whenever a vice president had succeeded to the presidency or had died or resigned from office, the vice presidency remained vacant until the next presidential and vice presidential terms began.