Well if you bypass the heater core youre heater will not function... at all. Not a good idea for winters.
Youll need to replace it. I replaced my heater core about 4-5 months ago with a new one from autozone. $30 I think. The hard part is removing the dash. EVERYTHING has to come out.... EVERYTHING! Youll be left with a steering wheel and a firewall when youre done. It is not fun, but with teh help of a buddy you can have it done in a weekend. I had mine out and back in within a day... but it was a very very long day.
You could always pay a shop to do it... but at $60-$90 an hour for this job... its gonna get steep REAL fast! most places around me charged at least $800 for the job.
Normally I say go aftermarket, but unfortunately some things are best left to OEM. This is one of them. Case in point, after replacing my heater core in my Saturn last year it went bad again, this year one year later, so two leaking H/C's later from after market parts I finally went to the dealer, also at the same time the H/C went bad in my PT Crusier, so I went with an after market H/C that I thought was made in the USA, was made in China. I put it in and it stinks to hi heaven, so while I hate to tell anybody to get OEM, this is one of things I won't skimp on in the future. I don't think the after market parts in this case are any better then OEM, I actually think they are worse in some cases. And I really don't want to do this job for practice, I do enough of that at my real day job.