Yeah, you can, just get a replacement cord at lowes with the cords already cut off and wires exposed.
Cut the ends off the UV and basking bulbs, wire positive to positive (white) and negative to negative (black) and if there is no ground needed, cut it off.
Replacement Cord
http://www.lowes.com/lowes/lkn?action=productDetail&productId=102225-353-95104
(THERE ARE CHEAPER ONES IN THE STORES, LIKE $10) DON'T BUT THIS ONE, JUST NEED A "REPLACEMENT CORD".
Buy that.
Cut the ends off the UV bulb cord and cut the ends off the basking bulb cord, expose the wires.
Connect the basking bulb wire positive and the UV wire positive to the positive (white) on the replacement cord.
Then
Connect the basking bulb wire negative and the UV wire negative to the negative (black) on the replacement cord)
Cut the green wire off, It's the ground, you don't need it for this.
If you have a solder gun, solder the wires together, if you don't just make sure they are good and wired together.
Get some shrink tube to cover the exposed wires.