O'Donoghue, who was born in London, grew up in Brisbane but is eligible for Ireland, has already started training with the Adams Park outfit and could feature against London Irish on Saturday.
The 29-year-old second-row previously spent two years with Ulster and has also represented the Queensland Reds and Northampton.
O'Donoghue earned international recognition for Ireland 'A' in February 2009 against Scotland 'A' and was included in Ireland's squad for a tour to New Zealand and Australia.
He made his first Ireland appearance in an uncapped defeat to the Barbarians in Thomond Park and was also selected for a game against the New Zealand Maori.
"Wasps is a great club with a lot of history. With a relatively new group of players and coaching staff, this is a great opportunity for me to be part of a club moving in a good direction," said O'Donoghue.
Wasps director of rugby Dai Young said: "Ed has got a lot of experience in some quality rugby environments and he will be a good addition to the squad."
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