How can I solder to stainless-steel semi-rigid coaxial cable?
Unfortunately, I don't have the tools that are needed for spot-welding or for proper crimping (Anindo's answer), and I fear that the dielectric of the coaxial cable won't survive the temperatures that are needed for brazing (Brian's answer). The solution that worked for me was using a specialized flux, as Scott and RedGrittyBrick suggested.  

For future reference, these were my attempts at soldering to the surface of the stainless steel shield of the coax cable:


PbSn solder with a flux core (standard electronics solder): The solder did not wet the surface at all, and consequently did not stick.  Sanding the surface with very fine sandpaper did not help.
Applying a more agressive F-SW 21 flux (3.1.1, brand "Lavar") that I found in our workshop A tiny bit of solder stuck, but not a lot. Surface was cleaned beforehand with acetone, then sanded with very fine sand paper.
Wrapping an (uncoated) copper wire around the stainless steel cable, then covering it in standard electronics solder: This created a surprisingly good mechanical/electrical joint. In the picture it is shown on the left. You can see that the solder does not wet the stainless steel surface, so the connection is probably made by the copper wire sitting on the stainless steel surface. One experimenter claims to have used such a joint without problems for years, but I doubt it is very stable over time, as oxygen can sneak in and corrode the contact.
Dissolve zinc in H-Cl to create ZnCl flux. I used 10-20ml of 37.5% hydrochloric acid and dropped a few tiny pieces of Zinc in it (less than 1 g), and waited 30 minutes until it stopped bubbling (well, actually there were still a few bubbles left). This worked as a flux, but I needed to put a lot of it onto the stainless steel surface until the solder started to stick nicely. This flux led to a lot of discoloration during the soldering (see middle section of the cable on the picture). Maybe it would have worked better had I cleaned the surface or sanded it down before. Note to self: When your local chemist recommends using a fume hood, then use a fume hood.
Using special stainless-steel flux: I managed to get a small sample of BrazeTec Soldaflux Z, a strong, but corosive 3.1.1a flux. This worked great, even without any prior surface treatment! Only a tiny amount of flux is needed. There is a nice, smooth transition between solder and the stainless steel surface (see right side in the picture). The downside: The flux expires after 6 months and can only be ordered in 1 kg amounts (ca. 70 €). 




It is sometimes claimed that the acid in the aggressive fluxes, together with moisture, carbon dioxide, and time, will lead  to corrosion of the lead of standard electronics solder. Maybe using lead-free solder, such as SN96.5 AG3 CU0.5, is a better idea.