Hi Ernie, thanks for the info. I must say, it's a little complicated. For the time being, my Gmail setup as an outgoing server works. Thank heaven's for that.... it might take me three weeks to get through that article! 🙂
Mick, thanks again for your input. That gave me the courage to go and talk to our computer man. I would have done this before, but he's usually swamped with all his different hats, and he knows nothing about the Mac. And, I must also add, I haven't learned enough German (I'm in the German part of Switzerland) to be able to communicate quickly. But, I did go talk to him this morning because you said that your IT guys wiggled their fingers and now it works. Come to find out, here's what happened... or well, never happened:
Around the time that this problem started, we upgraded our internet connection to a higher package... what I mean is increased bandwidth/speed. Nothing on the side of hardware was changed. The ISP only provided us with more speed/bandwidth. But, what I did discover is that the error that I found in the connection doctor is a relay problem. Our dear network brother tried his, using the same smtp that I have and he got the same problem: relaying denied. Go figure. He doesn't know what that means, but that problem showed up also for them along with the speed upgrade. This is exciting new information. Of course it doesn't fix the problem nor tell us what's really wrong. But it definitely shows that it's not a Mac problem. Good ole Mac! It does "just work".
So, through our own mail server (which I learned that we have) I'm able to use the smtp. This is better because can I can send from my original e-mail account with our smtp and it will show the correct e-mail. Of course I can still send with gmail, but the recipient sees my Gmail address and not the correct one. I hope I explained that in an intelligible way.
Perhaps Apple would do better to actually put the error message that the connection doctor can see in their dialogue box instead of just saying that mail can't send the message. I don't know.