There are a number of things to try.
What my spouse did:
Bars with a 30 degree rise, in lieu of the flat bars.
Taller stem (his bike has a quill stem so it was easy to change out. If you have a threadless stem you will need a threadless stem extender)
Ergon Grips:
Even this is not a 100% solution.
Also, I suggest doing core strengthening exercises. The stronger your core the less weight may end up on your hands. Here are some to try:
http://www.mayoclinic.com/health/core-strength/SM00047
Some people like to use bar ends to increase the number of hand positions. Both spouse and I tried them out and didn't find them helpful.
On one of my bikes I have flat bars with Ergon drips and they are pretty good. On another I put Sparrow bars. This puts my wrists in a neutral position. These bars are comfortable for rides of up to about 50 miles, even sans gloves.
The Sparrow bars:
