Amendment XVII
The Senate of the United States shall be composed of two Senators from each state, elected by the people thereof, for six years; and each Senator shall have one vote. The electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislatures.
When vacancies happen in the representation of any state in the Senate, the executive authority of such state shall issue writs of election to fill such vacancies: Provided, that the legislature of any state may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.
This amendment shall not be so construed as to affect the election or term of any Senator chosen before it becomes valid as part of the Constitution. "
(Before that amendment, the COTUS stated that state legislatures chose the Senators from that State.)
BUT,
Article I, Section 5 of the COTUS states in pertinent part:
"Section 5. Each House shall be the judge of the elections, returns and qualifications of its own members, and a majority of each shall constitute a quorum to do business; but a smaller number may adjourn from day to day, and may be authorized to compel the attendance of absent members, in such manner, and under such penalties as each House may provide.
Each House may determine the rules of its proceedings, punish its members for disorderly behavior, and, with the concurrence of two thirds, expel a member."
In the 1969 case of Powell v. McCormack, though, the SCOTUS interpreted the analogous powers of the House to qualify duly elected Representatives rather narrowly.
http://caselaw.lp.findlaw.com/scripts/getcase.pl?court=us&vol=395&invol=486