Tentative good news this time. While the site is still flagged for malware, the report is that on the latest review the site was found clean, and the malware warnings are being removed.

The last thing the hosting provider said they did was to recompile the Apache server, so, if this sticks, that will be what did the trick (they also earlier made some unspecified modifications to security settings).

Because I understand this kind of problem may be intermittent and return, I'll be looking at it in Google Webmaster Tools twice a day, so I can be right on top of the issue if the problem returns. But, knock on wood, so far so good. If you don't hear further from me, it will be good news.

The other site from my hosting provider that had reported a similar problem looks to now be free of malware warnings in Google search.

Thanks to everyone for your help.