Yes, you can do this without reinstalling if the drive that the rest of the linux installation is on will remain in the box. It's even easier if you can leave the old drive in (to use the old swap partition) while you create the new partition.
If you can't leave the old drive in, I would first comment out all references to the old drive that you are removing in /etc/fstab, including the line that mounts the current swap partition.
Then shutdown, remove the old drive, install the newdrive and startup. disclaimer: I have never started linux without a swap partition mounted so i don't know if it will complain when you boot up. I would suggest booting into 'single' mode to minimize the services that you start (and minimize memory usage) while you partition the new drive.
Use cfdisk to partition the drive and create a 'linux swap' partition. note the partition name that the swap will reside on (something like hdb2).
After partitioning, run
> mkswap /dev/hdb2
(or whatever the partition name is) You can use the -c flag with mkswap and it will check the partition for bad blocks first.
Then edit fstab and replace the old partition name with the new /dev/hdb2 and uncomment the line.
then run
> swapon -a
you can do
> cat /proc/meminfo
to check that the new swap partition is recognized.
Good Luck.
jaa