On February 12, 1937, it was announced that the Cleveland Rams would be leaving the American Football League to join the NFL. One year later they would take the place of the St. Louis Gunners in the Western division. The Rams stint in Cleveland was short. After the Championship season of 1945, the team packed its bags and headed for Los Angeles. The team resided in L. A. for 48 years before shifting to St. Louis in 1995.
This was not the same AFL that merged with the NFL in the late 60's. The AFL the Rams were originally a part of, was a league that lasted just two seasons, before giving way to the league that has proved to have true staying power as time has progressed. Cities such as Boston, Minneapolis, and Buffalo all had hopes of being the newest addition to the NFL, but the financial stability that the owner of the Rams, Howard Marshman brought to the table won out in the end.
It was not an easy transition for the franchise to make. Between 1937 and 1942, their best finish was third place, and even then their record was below .500, as they went 5-6 in '42. World War II nearly spelled the end of the franchise. The entire '43 season was scrapped because of a player shortage as many of the young men chose to serve their country, rather than play a game. but came back the next season. The Rams were back on the field in '44, then finally saw a complete turnaround in '45 with the emergence of a rookie quarterback that was destined for the Hall of Fame. That quarterback was Bob Waterfield, he led them to a 9-1 record before winning the NFL Championship game 15-16 over the Washington Redskins. That title was secured by an early safety. Less than a month after winning the title, the then owner of the Rams, Dan Reeves, announced that they were headed west. It must have been a bittersweet time in Cleveland to watch the team go, after they had finally achieved great success.
You can read about the 1945 title game here: http://onthisdayinsports.blogspot.com/2013/12/december-15-1945-early-safety-leads-to.html