Well, I have done a bit more research, and here's what I've found:
There are two versions of JBMail.exe available at the above site - an .exe installer, and a .zip archive. I would have expected that both would have the same files, but the .exe installer installs an uninstaller program, and this apparently is what actually triggers the false positive response. Downloading the .exe installer causes Avast to complain immediately. Extract the files or run the installer, and it is the file "uninstal.exe" that Avast reports as malware.
The direct link to the file reported as malware:
http://www.pc-tools.net/files/win32/trialware/jbmail32.exeAt
http://virusscan.jotti.org, only Avast reports this file as malware.
At
http://www.virustotal.com, two other scanners (eSafe, Fortinet) report it as "suspicious", but only Avast is sure that it is malware.
This installer is several years old and has not been recently modified. I have used it many times on many different machines and never seen any adware installed as a result. I am quite thoroughly convinced that it is very safe, and that these malware reports are false positives.
Also, this same file has been on my computer for years (as has Avast), but this is the first time that Avast has ever complained about it. This leads me to believe that it is a recent change to the database that is causing these false positives.