My car is a 2001 zx3 i don't know if is the same on a 2003 .. but here's how i did it..
1. Remove the dome light with a tiny flat screwdriver
2. After have the dome light pull out alittle with screw driver just pull it out by hand
3. Now attach the wire on the bulb like picture below- just remember which wire you connected. I pinch the wire using the same tiny flat screw drivers.
This color is from the wire i use to connect the cold cathode..
Possitive = Gray
Negative = Gray/ Black Stripe
Here's a link for the 3rd picture in 1600 x 1200 you can zoom in more..
http://www.turbozx3.com/files/domelamp_3b.jpg
Hope this helps.
PRTurboZX3