Venting a direct vent fireplace into an existing wood burning stove flue
Hard to say without seeing it, but typically you can use the existing flue provided you add the proper direct vent chimney insert for your particular gas fireplace. Alternatively, you can possibly knock a hole in the back of your chimney and just go out the side if you'd prefer (more labor, but less flue inserts you'd need to purchase). 

UPDATE:

We actually did this recently. We ended up taking a direct vent unit and have the exhaust going through the existing flue (about 20'). For intake air, we're letting it take it from inside the house (since this isn't a primary heating unit, the loss in efficiency isn't a big concern for us). We've ran it a few times and seems to be working just fine.