Also, if the would-be successor does not meet the constitutional requirements for President then the presidency would skip them and go to the next person in the list.
1) Vice President
2) Speaker of the House of Representatives
3) President Pro Tempore of the Senate
4) Secretary of State
5) Secretary of the Treasury
6) Secretary of Defense
7) Attorney General (aka: Secretary of Justice)
8) Secretary of the Interior
9) Secretary of Agriculture
10) Secretary of Commerce
11) Secretary of Labor
12) Secretary of Health and Human Services
13) Secretary of Housing and Urban Development
14) Secretary of Transportation
15) Secretary of Energy
16) Secretary of Education
17) Secretary of Veteran Affairs (thanks CentrX and narzos)
18) Secretary of Homeland Security