The 34-year-old rapper from Gary, Indiana has withdrawn his appeal against a French court's extradition order handed down in late June.
Gibbs was arrested on June 2 shortly before a planned concert in the southwestern French city of Toulouse, one of a string of European tour dates.
Gibbs, whose real name is Fredrick Tipton, was the subject of a European arrest warrant issued on May 31 for an alleged rape in Austria last year.
According to Austrian media, citing prosecution sources, Gibbs is accused of drugging two women who came backstage after a concert in Vienna last July, before taking them back to a hotel where they were allegedly raped by the rapper and one of his security agents.
The singer, who denies the rape allegations, initially announced his intention to appeal against extradition, but has changed his mind, said his Toulouse lawyer Michael Malka.
"The decision of the Toulouse appeal court has therefore been able to be executed, with a transfer to Vienna on July 29," he told AFP.
While he was free to live in Toulouse after paying 50,000 euros in bail, Gibbs "voluntarily withdrew his procedural extradition appeal," said his US attorney Theodore Simon.
"He did so to work cooperatively with authorities to assure a smooth transfer to Austria and to further enhance the opportunity for an expedited decision on the merits," he added in a statement.
In Austria he will "ask the court to release him on bail, under reasonable terms and conditions, as was done in France," it added.
"He continues to pursue every available means to demonstrate that he is absolutely innocent and that he has been both belatedly and wrongly accused.
"To be clear and accurate, to date Freddie Gibbs has not, we reiterate, Freddie Gibbs has not been charged with any offence."
While an investigation was continuing, Gibbs "remains hopeful that a thorough and searching investigation will reveal the actual facts, including the absence of any scientific, physical, or credible evidence implicating him, thereby paving the way for his exoneration and return to his family and one-year old child."