Who becomes president after the death of the vice-president?

If the US president dies, the vice-president becomes president and a new vice-president is appointed.

If the 'new' president dies, the 'new' vice-president becomes president.

This process is dealt with under the 25th Amendment to the US Constitution which starts off....

Section 1. In case of the removal of the President from office or of his death or resignation, the Vice President shall become President. Section 2. 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.