I tried a lot of different things. First, I tried downgrading ZoneAlarm, I tried upgrading Shadow's Experimental BitTorrent client to the latest version (which is called BitTornado), I tried closing every program except ZoneAlarm and BitTornado, I tried Azureus -- all to no avail.
I asked around, and breyten told me he had troubles with BitTorrent and ZoneAlarm in the past as well -- though kees_s had it all working without a hitch. breyten told me he didn't use a firewall anymore because he is behind a NAT, but he should know better: NAT is not a firewall! I don't feel comfortable broadcasting my IP adress to just about anyone who connects to the BitTorrent tracker without being behind a firewall...
So, the hunt was up for another firewall! I went with the first I came across that looked trustworthy: the Kerio Personal Firewall. The interface is surprisingly similar to ZoneAlarm, with one added advantage: it doesn't crash under the barrage (torrent?) of packages when using BitTorrent.
I've let Azureus run full-speed during the night, and this morning, everything was still working perfectly! So goodbye ZoneAlarm, hello Kerio!