Thanks for the responses. That Cardas kit looks very nice. The pot is a lot larger than mine though, I wouldn't want to fill that one with fluxed solder!
I tried to get some core-less solder at the best electronics shop in town, but no luck. So I settled for a bit of the included solder to try out. I melted the solder in the bath, but with John's remark in mind I did it outside. I was glad I did! It produced lot of smoke while the flux burned away, but then I could take it inside without a problem.
First results were mixed: the wire was often dirty when it came out of the solder, or the solder wouldn't hold. So I poured the solder into a cup of water (the solid droplets can be easily cleaned and look quite nice by the way) and cleaned the bath.
Tonight I tried again and got it working very nicely: a bit of flux on the wire, then a few second dip (until the foam/bubbles of boiling varnish could be seen at the surface). Take it out vertically and no dirt will cling to the wire (the surface is dirty with burned residue, but the boiling action pushes the dirt away). The wire end is covered with a nice, even layer of solder. Great!
Now all I need to do is to put in more solder (the bath is not deep enough yet). Glad to hear that regular solder will do, I have more than enough of that.