Brown, 28, the team's third-down receiver in recent years, will compete for a starting job now that Shawn Jefferson has gone to Atlanta.
Brown's five-year contract is estimated at $13 million, with a signing bonus of around $3 million, The Boston Sunday Globe reported.
"This is what I wanted all along," Brown told the newspaper. "It was an interesting process and I would have gone to visit some other cities, but the Patriots finally put what I needed on the table. I'm very excited this could happen. Now I can prepare to stay with the team that drafted me."
Brown had visited Kansas City and scheduled a visit with the Cleveland Browns.
Brown caught 36 passes last year for 471 yards, averaging 13.1 yards per catch even though he missed four games with an ankle injury. He averaged 10.7 yards per punt return, with long returns of 52, 32, and 26 yards. He also returned eight kickoffs for a 33.9-yard average.
He had rejected a previous contract offer of $2.5 million per year because he felt too much of his pay would have come toward the end of the contract. But the Globe reported that the Patriots then put more of the money up front in the signing bonus and more evenly distributed salaries earlier in the contract.
Jefferson was lost to free agency two weeks ago when he signed a five-year, $18 million deal with a $4.5 million signing bonus with the Falcons.
Bjornson caught 10 passes for 131 yards and one touchdown for the Cowboys last season. But in 1996 and 1997, his best seasons, he caught 48 and 47 passes, respectively.
The Patriots have been looking for depth at tight end after they cut Ben Coates for salary-cap reasons.
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