Leiua has played 30 games for Wasps since joining them from Super Rugby side Hurricanes in April 2014.
The 28-year-old, who can also play on the wing, has scored two tries in 11 Test appearances for Samoa.
"He's a robust midfielder who can play in a number of positions in the backline," Bristol head coach Mark Tainton told the club website.
"Alapati was a standout performer for the Hurricanes in Super Rugby and made a similar impact in his first season with Wasps - he will be a fantastic addition to the Bristol Rugby squad."
Bristol are currently bottom of the Premiership table, one point behind 11th-placed Worcester with nine games left to play this season.

Theresa Villiers and Charlie Flanagan both attended a meeting to review the implementation of the Fresh Start deal negotiated in November last year.
Both the British and Irish governments said the momentum created by the agreement must be maintained.
Mrs Foster became Northern Ireland's first female first minister on Monday.
The Democratic Unionist Party leader took over from Peter Robinson, who confirmed his resignation in the Northern Ireland Assembly.
Mrs Foster is the youngest person to hold the most senior post in Northern Ireland politics.
Speaking after Thursday's meeting, Mrs Foster said she was "absolutely delighted" by how the Fresh Start agreement had been implemented.
"Good progress has been made and we'll continue to make that progress," she added.
She said the legacy of Northern Ireland's Troubles had been discussed and that there was a need to talk to victims' groups.
Earlier, Northern Ireland Secretary Theresa Villiers said "significant progress" had been made since the Fresh Start agreement was made, but "substantial challenges" remained.
She added: "Work continues with victims' representatives and others on finding a way forward to address the legacy of Northern Ireland's past."
Bristol Rugby have signed Wasps' Samoa international centre Alapati Leiua in a deal to start from next season.
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Arlene Foster has had her first meetings since becoming first minister with the Northern Ireland secretary and the Irish foreign minister.