According to Cornell University Law School, the 25th Amendment deals with presidential succession. It states in Section 1: "In case of the removal of the President from office or of his death or resignation, the Vice President shall become President."
Cornell's printing of the Constitution reveals the various sections of the 25th Amendment, which detail the various ways in which succession happens. Section 3 of the amendment states that if the president declares in writing an inability to perform the duties of his office to the president pro tempore of the Senate and to the Speaker of the House, the vice president becomes the acting president until the president regains his abilities and takes back the office with a written notice. Section 2 notes that if there is a vacancy in the position of vice president, the president nominates one. The nominee must be approved by Congress before taking office.