Yep, you've got a grounding problem. No volume pot, huh? First thing one normally does is...go back and make sure your solder joints are good on the jack. I know, I know, you've checked 'em already. Do it one more time, very carefully, just to be sure. You'd be surprised at what you can miss the first two times around. Next, check your wiring at the bridge and the jack. How did you attach the wire to the bridge? Brass bridge, eh? Hmm. It should be just conductive enough.What about attachment screws? Try running a wire from the screw, instead of the bridge, to the negative lug on the jack.
Forgot something; try twisting the leads from the pup into a braid. Make sure you connect + and - to the appropriate lugs on the jack. It won't cure the source of the hum, which is definitely a grounding problem, but it will provide cheap shielding to prevent other problems after you solve the grounding issue.