druuna and syg00, you both beat me to the punch. Man, I just type too slow.
knipknup, you'll want to do as syg00 suggested, and reduce your swap to a size that is reasonable. The logical partition(s) you create will then start at /dev/sdb6 and will be within that extended partition along with your /dev/sdb5 swap.
This is just a matter of logistics, but with all of these types of disk resizing exercises, the devil is in the detail.
Look at your /etc/fstab file to see how your swap is defined. If it looks like this,
/dev/sdb5 swap swap defaults 0 0
you are all set to just create another appropriately sized /dev/sdb5 swap partition.
sudo swapoff /dev/sdb5
sudo fdisk /dev/sdb
sudo mkswap /dev/sdb5
sudo swapon -a
'swapon -s' will show you it is swapped on properly.
[tommy@athlon ~]$ swapon -s
Filename Type Size Used Priority
/dev/dm-1 partition 6291452 0 -1
If your /etc/fstab entry for the swap partition starts with UUID= or LABEL=, use that information and either the '-L' or '-U' flags of 'mkswap' so that your swap file and fstab entry match.
I don't know Mint and don't know if mounts and swapon's are done by LABEL or UUID (yet), but many distros are moving in that direction. It doesn't hurt to look. If the entry in fstab is wrong the worst that can happen is that no swap gets mounted ("swapped on") at boot time - a failed swapon doesn't prevent the system from coming up.
And just a question from me to all of the Mint literate out there, or for you, knipknup: where is /dev/sda? Just curious.