Are you sure it doesn't have some sort of temp control? Seems to me it would have one someplace - maybe hidden around the back near the compressor?
If so and if you are lucky, may be just a defective controller which would a cheaper problem to resolve than a compressor. If you culdn't get a new one, you could bypass it and buy an external controller to switch the fridge on/off (as I have on my kegerator) for around $50.
Just a thought - good luck with it anyway.
Brian