The Royal & Ancient organisers said in a news release on Monday that five Race to Dubai money-list exemptions had been secured by Bjorn, South African Richard Sterne, Australian Brett Rumford, Finn Mikko Ilonen and Briton Marc Warren.
The 42-year-old Bjorn will be playing in his 16th Open. He has twice finished tied second, at St Andrews in 2000 and at Sandwich three years later.
The Dane was also a double European Tour runner-up last month, at the BMW International in Germany and the Austrian Open.
Former Ryder Cup player Weekley, Billy Horschel, Russell Henley, Jimmy Walker and Harris English have booked their tickets for the third Major championship of the season through the US PGA Tour's FedExCup points table.
Weekley ended a five-year drought by winning the Crowne Plaza Invitational in Texas in May.
He finished tied 13th at the Open in 2009.
Swede Jonas Blixt, winner of the Greenbrier Classic in West Virginia on Sunday, is among eight players to get into golf's oldest Major through the world rankings.
The other seven are fellow Swede Fredrik Jacobson, British pair Chris Wood and Martin Laird, Americans Ken Duke and Kyle Stanley, Marc Leishman of Australia and Graham DeLaet of Canada.
Golf's oldest Major championship begins on July 18.
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