Mr Burris swears that he offered nothing in return for the seat
Democratic senators have endorsed Roland Burris, the man selected to fill Barack Obama's vacant seat.
Senate leaders initially tried to block Mr Burris's membership, because he was appointed by scandal-hit Illinois Governor Rod Blagojevich.
Mr Burris said he was "humbled and honoured" to be Illinois' next junior senator, after senate leaders relented.
Mr Blagojevich was arrested last month and charged with attempting to "sell" the vacant seat to the highest bidder.
He has defied calls to resign, and also calls to delay naming a new senator.
Under Illinois law, the state's governor has sole responsibility to fill vacancies in the senate.
Mr Burris was initially turned away from Congress when he attempted to attend the swearing-in last Tuesday.
Officials said he did not have the correct credentials because his appointment had not been signed off by the Illinois secretary of state.
But after discussions with Mr Burris, Senate majority leader Harry Reid and senior Illinois Senator Dick Durbin said he had provided fresh paperwork confirming his appointment.
"We have spoken to Mr Burris to let him know that he is now the Senator-designate from Illinois and as such, will be accorded all the rights and privileges of a Senator-elect," the two Democrats said in a statement on Monday.
They added that they expected Mr Burris to be sworn in and formally seated later this week, barring objections from Republicans.
On Friday, the Illinois House of Representatives voted 114-1 to impeach Mr Blagojevich.
Illinois's state Senate will now try Mr Blagojevich. If found guilty, the governor will be forced out of office.
He was arrested in December and charged with three counts of soliciting bribes, and offering to distribute public funds in return for political favours and campaign contributions.