No thanks to Apple, but kudos to Adrian232 on macrumors forum, I have the fix! Such a shame that Apple couldn't or didn't care to fix it.
Here is Adrian's fix:
Basically, the fix is to put all your mail passwords in a separate, always-unlocked keychain.
Here is how to do this:
1) Open keychain access, go to File -> New Keychain. I named mine Mail. Save it in your
keychains folder (default), and give it a password.
2) Right click the new keychain and select "Change Settings for Keychain". Here, uncheck the "Lock after" and the "Lock when sleeping". The latter can be left on if you don't mind re-typing your password every time your computer goes to sleep (can also be more secure if you don't lock your whole machine when sleeping).
3) Here is the tricky part: you need to find all your mail keys and drag them into the new keychain. Do a search for your mail host or your login username and you might see a list of keys. It will be flagged as an "Internet password", not web form, and when you click on it it should read "imap://your.web.host.tld/" or pop if you use that, and "smtp://your.web.host.tld/" for your outgoing mail. Just drag them all into your Mail keychain, type in your password and click Accept. The keys will stay on the search screen, but clicking on the Mail keychain will confirm that they are in fact moved.
4) Now lock your primary keychain (the one in bold) and run a test by checking your mail. If no dialog pops up asking for your keychain password then it was a success!
I have now been nag free for the past 24 hours and it is grand! This solution is far more secure than simply keeping your primary keychain unlocked (for the reasons I specified in my last post) as all your other passwords from html forms, etc are still locked but your email passwords remain unlocked in Mail all the time. Remember, no human can get at your actual passwords in your keychain except you.
I believe I followed the instructions. I made a new keychain called Mail and gave it a password. I then came to the "tricky part" and looked for mail related items in the other keychains. I dragged them to the new Mail keychain. I unlocked the padlock on "Mail" and tested the Get Mail button, but got the Password demand. I tried unlocking the other keychains and again got the Password demand! Maybe I missed something. With all the keychains unlocked, how does the problem persist?
I suspect that the best solution is to abandon Mail and use the BT email app instead.
I only hope I have not screwed up something else!
Going through your procedure somehow erased the password in my mail account, so that any request for mail resulted in the box requesting the password.
Entering the password seemed to solve the problem, but now this morning the password request came up spontaneously as before when I was not using Mail. Like you it seemed to work for a while.
Even turning to another email programme will not stop these apparently random requests for the password.
I do not think this is a problem with how we are using the keychain, I think it is a bug that only Apple can fix.
How do we report it to Apple?
IN CASE APPLE ARE LOOKING AT ALL THIS - COME ON APPLE SORT IT OUT PLEASE.
I would like to find an alternative to Apple Mail which continues to show the same problem
My BT broadband subscription uses Yahoo, but is dogged with stupid adverts, although otherwise OK.
Does Mozilla Thunderbird carry adverts?
If not how is it paid for?
Does it have any other problems? I would like to be sure before I swap.
I have this problem, as well. I have 7 email accounts in apple mail. Three of them frequently ask for my password. They are all with the same isp, however, I have 2 other accounts with this isp & I am never asked for a password.
Afew months ago, I upgraded to snow leopard & thought the problem was fixed. About a month ago, it started again! Frustrating! It does make me consider another mail program.
Please be aware that Apple provides these forums for end users to help other end users. This is not a Apple supported support site. Wether they reads these posts or not, do not expect support from Apple at these forums.
Rather than venting send feedback to: http://www.apple.com/feedback/ Team Mail will read those post but not these in the forum. These forums are not unlike other forums run by private persons that says here is a site to help others.
I am running Leopard as you state you are and started having same problem. I WAS trying to access my email from Yahoo using POP accoun, using MAC mail. I found the issue was YAHOO not MAC Mail! They have a known issue since March 4, 2012. I tried using Yahoo Live chat twice and neither tech agent had a clue. I contacted my ISP to make sure they were not blocking any ports MAC Mail was using and they were not. I then paid for Apple Support and two different advanced techs worked on it with me and the second agent finally found on YAHOO web sight the issue listed. Yahoo says they are working on a resolution, I refuse to wait any longer dealing with Yahoo and their Un-knowing tech support.
I switched to Gmail, set up imap, and have absolutely no further problems!!!!!!!!!!
Check and make sure you can log into your mail account from a web browser like Safari, so you know the problem is NOT your password.
I am thankful the Apple Care agents did not give up and were the ones that found the problem with Yahoo. I am now an other faithful user of MAC. I will never use Yahoo for anything.
I forgot to mention in my other response I deleted ANY and ALL of my keychains saved for my Yahoo log in. I had a lot as i always checked Save in Keychain each time i entered my password. After setting up the new Google account I did not enter any keychan login passwords except where prompted in the Add New Account forms and have not been asked for password since.