Burton says his agent, Todd France, finished the deal on Thursday. Burton says he believes the contract to be worth more than $1 million.
Burton was selected by the Rams in the fourth round of the NFL Draft in April.
Burton ranks fourth in receptions, third in receiving yards, second in touchdowns and third in all-purpose yards for the Wildcats. He was the fifth Kentucky player to amass more than 4,000 all-purpose yards.
Burton's best season came in 2006 when he caught 77 passes for 1,036 yards and 12 touchdowns.
Burton will report to training camp July 25.