What should I do with my leaking chimney?
Ventless gas fireplaces are a suspected health hazard, and may dealers won't sell them.

I strongly suggested a vented model, which will have two pipes: one for fresh air, one for exhaust.  The new cap will likely solve your dripping problem:



If you have doubts about where the leak comes from, get a hose.  Wet the chimney 1 meter above the ground and wait for the drip to stop.  Repeat this until you get to the top of the chimney.  I'll bet the leak is in your chimney cap, or whatever masonry topping the thing has.