Constitutional FAQ Answer #66
Q66. "If the President dies and then the Vice President takes over and dies too, who takes over then?"
The Vice President. One of the VP's first acts as a new President would likely be the appointment of a new VP. The new VP must be confirmed by both Houses of the Congress (viz 25th Amendment). If both die at the same time, the Speaker of the House is President. The entire line of succession is spelled out in the U.S. Code
(and elsewhere in this FAQ).