The answer is most likely no... unless you have a Serial ATA hard drive (which uses a very thin cable that looks nothing like the flatter and wider IDE cable), then your hard drives do indeed share the SAME ATA ribbon cable as the DVD/CD ROM drives!
If your current setup is as you described, you are using THREE of the possible FOUR connections on your IDE ribbon cables (two IDE HDs and one IDE DVD drive).
Most IDE Ribbon cables have three connectors on the cable... one on each end and one in the middle. Some cables have only the connectors on each end and nothing in the middle. In any case, all IDE connectors on your motherboard (of which there should always be TWO of them on the MB) can support two devices each... which totals FOUR IDE drives (any combination of hard drives and CD/DVD drives).
All you need to do is connect your new DVD Burner to the middle connector on the ribbon cable that goes to your current drive and be done with it (be sure to have the correct master/slave jumper settings or use cable select on the drives themselves).
If for any reason, you got "gyped" and have an IDE ribbon cable with only the connectors on each end (and none in the middle) simply purchase a proper ribbon cable that has all three connectors on it!
That should fix you up!