On February 24th, 1949, a two stage V-2 'Bumper WAC' launched from White Sands, New Mexico became the first documented rocket to reach space. It attained an altitude of 244 miles, well above the currently accepted definition of space - 62 miles. Read more on Wired.
(Wikipedia notes that a German V-2 had reached 117 miles at some time in 1944. I don't know how reliable this figure is. The same article says the Bumper WAC's altitude was 248 miles.)
Update: Based on other discussions, I am convinced that this was not the first V2 to break 62 miles. Oh well.