This is now solved.

Over on IRC, cgminer author Con Kolivas himself (!) kindly suggested using AMD SDK version 2.7 and the AMD Catalyst Driver version 12.8. Both can be found on the AMD downloads area. This worked. Following the cgminer "README" is essential.

I experienced two wrinkles.

1. Installing AMD's driver required first uninstalling the version of fglrx from the Ubuntu repositories (sudo aptitude purge fglrx, sudo service gdm restart [or just reboot], sudo rmmod fglrx).

2. The script "Install-AMD-APP.sh" supplied by AMD in their package didn't seem to do everything that was actually required. I followed the instructions in the cgminer README describing how to install SDK version 2.4, but changing "2.4" to "2.7". After this step, it was possible to compile and run cgminer.