I think I have solved the problem (thanks to Roger's help)
I don't know if it's always been like this, but SSL was turned off - I turned it on and it seems that I can now send emails without any trouble.
To fix it I did the following:-
Open Mail => Preferences… => Accounts
Select my account from the "Accounts" column
Select "Account Information" from the tabs in the tabs pane
Click on the "Otugoing Mail Server (SMTP):" drop-down menu and select "Edit SMTP Server List…"
Select the "smtp.me.com" server from the "Server Name" column
Select Advanced from the tabs below
Click "Use Secure Sockets Layer (SSL)" to turn it on (it was off in my case)
Note:-
At this point I happened to have the Mail Connection Doctor turned on (Open Wndow => Connection Doctor) and I noticed that if I turned SSL off the connection would fail and if I turned SSL on, the connection would succeed.
Click OK to close the "Edit SMTP Server List…" pane
Close the Mail Preferences pane
Additional note:-
It appears that I did not have to change "smtp.me.com" to "smtp.mail.me.com" as Roger suggested - I'm not sure what relevance this has.
Again, thanks to Roger Wilmut1 for his help.
Mark T