DNS on static IP address
Solution 1:
As a final resort I disabled and then enabled my networking. This solved the problem. It appears that the connection manager can't apply the connection settings until the connection is restarted.
The user interface gives no indication that the settings are not yet applied. Which is really quite poor.
Solution 2:
I don't know how to do it via the gui, but if you type:
echo nameserver 192.168.1.1 >> /etc/resolv.conf
as 'root' that should work for you (if you aren't root, you'll need to through a sudo in front of that).