The Presidential Succession Act of 1947 (3 U.S.C. § 19) establishes the order of succession to the Presidency in the event neither a President nor Vice President is able to "discharge the powers and duties of the office."
The authority for Congress to do this is in Article II, section 1, clause 6 of the Constitution: "In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the same shall devolve on the Vice President, and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected." (Modified somewhat by Amendments XX and XXV.)
The order of succession:
Speaker of the House
President Pro Tempore of the Senate
Secretary of State
Secretary of the Treasury
Secretary of Defense
Attorney General
Secretary of the Interior
Secretary of Agriculture
Secretary of Commerce
Secretary of Labor
Secretary of Health and Human Services
Secretary of Housing and Urban Development
Secretary of Transportation
Secretary of Energy
Secretary of Education
Secretary of Veterans Affairs
From Wikipedia:
Constitutional concerns
Several constitutional law experts have raised questions as to the constitutionality of the provisions that the Speaker of the House and the President pro tempore of the Senate succeed to the Presidency.[1] James Madison, one of the authors of the Constitution, raised similar constitutional questions about the Presidential Succession Act of 1792 in a 1792 letter to Edmund Pendleton.[2] Two of these issues can be summarized:
The term "Officer" in the relevant clause of the Constitution is most plausibly interpreted to mean an "Officer of the United States", who must be a member of the Executive or Judicial Branch. The Speaker and the President pro tempore are not officers in this sense.
Under the principle of separation of powers, the Constitution specifically disallows legislative officials from also serving in the executive branch. For the Speaker or the President pro tempore to become President, they must resign their position, at which point they are no longer in the line of succession. This forms a constitutional paradox to some.
In 2003, the Continuity of Government Commission, a private nonpartisan think tank, suggested that the current law has "at least seven significant issues … that warrant attention," including:[3]
The reality that all figures in the current line of succession work and reside in the vicinity of Washington, D.C. In the event of a nuclear, chemical, or biological attack, it is possible, perhaps even likely, that everyone on the list would be killed or incapacitated.
Doubt (such as those expressed above by James Madison) that Congressional leaders are eligible to act as President.
A concern about the wisdom of including the President pro tempore in the line of succession as the "largely honorific post traditionally held by the longest-serving Senator of the majority party." For example, from January 20, 2001 to June 6, 2001, the President pro tempore was 98-year-old Strom Thurmond of South Carolina.
A concern that the current line of succession can force the presidency to abruptly switch parties mid-term, as the Speaker and the President Pro Tempore are not necessarily of the same party as the President.
A concern that the succession line is ordered by the dates of creation of the various executive departments, without regard to the skills or capacities of the persons serving as their Secretary.
The fact that, should a cabinet member begin to act as President, the law allows the House to elect a new Speaker (or the Senate, a new President pro tempore), who could in effect remove the cabinet member and assume the office him- or herself at any time.
The absence of a provision where a President is disabled and the Vice Presidency is vacant (for example, if an assassination attempt simultaneously wounded the President and killed the Vice President).
Add Secretary of Homeland Security to the bottom of the list -- the law provides that cabinet members are added in the order their departments were created. At the time Congress created DHS, there was some talk about revising the order -- in any crisis that takes out the president, VP, president pro tem and speaker at about the same time, you probably don't want Homeland Security at the bottom of the list -- but I don't know of there's been any move to change the statute.