@phenomlab It seems that I may have it fixed.
As I could log in using the old domain, I took out the Session cookie domain setting in the admin section. Then log in using the new domain started to work. Actually, both domains work now.
Earlier I changed that value from the old domain name to the new domain. But that didn't work. Then I saw a tip under the setting box that reads "Leave blank for default". So I made it blank. I assume that the default is the host name in the HTTP request?
I want to think you for your writing. That lead me to the right direction.