The Speaker of the House of Representatives is next in line for the presidency after the vice-president. ( See the related question for the complete line of succession. ) (Be aware that there is a procedure for expeditiously filling a vacancy that might occur in the office of Vice-president. The president and vice-president would have to die at almost the same time for this line of succession to apply.)
In the US there is a line of succession should a president be unable to serve due to illness or death. The next in line to serve as president is the Vice President. This was seen when President Nixon resigned and Vice president Ford became president. Should a vice president be unable to serve, then the Speaker of the House of Representatives becomes the new president.