See Spacewar was made in 1961, I think theres one earlier than that:
WIKIPEDIA!:
On
May 5,
1951, the
NIMROD computer was presented at the
Festival of Britain. Using a panel of lights for its display, it was designed exclusively to play the game of
NIM; this was the first instance of a digital computer designed specifically to play a game.
[4] NIMROD could play either the traditional or "reverse" form of the game.
SRC:
http://en.wikipedia.org/wiki/First_video_game
Thats assuming you don't count the one on a cathode ray tube, which you probably wouldn't since all it could do was a dot.
My argument for the NIMROD computer is that it was specifically designed to play the game NIM - the whole computer is actually a game, however it was programmed to run that game, hence it's a computer game, just not in the normal sense where it could do more than one.
So obviously I'm going 10 years before Spacewars and going with NIM, since it needed the exclusively built computer, and it's a game. Computer, Game. Not a Personal Computer game though, but it probably was the first actual computer game that was more than a dot on a ray tube