OS X :: Get Mail Process Activated But Search Fails

I am using OS X version 10.5.7. Mail version 3.6 (935/935.3) As of yesterday my Mail program has failed. The symptom is that in the left window under 'mailboxes" when the "get mail" button is selected, the process activates, But the search fails - as though there is no internet connection. None of the accounts receive their mail, and the circle containing the exclamation mark signal appear. (Note, there is email arriving at the ISP. I checked using the ISPs webmail facility. I've tried retyping the password in the Mail settings, it's done correctly, so are the other settings for the ISP, but no luck). Sends have been successful.

I was hoping to extract (or archive? whatever that means) the actual messages and settings, then reinstalling the mail application. However, it seems from reading some posts that it might be necessary to instal the whole OS X system afresh to achieve this result. Of further concern to me is that it might not be possible to separate the messages in the mail application from the application itself. I don't want to just backup the mail folder then reinstal it, as i suspect there is something wrong with that folder, so I would just be reintroducing the bug into the reinstalled OSX... which would defeat the purpose.