After some experimentation, I found I can get routing working
again - temporarily - by following these steps:

run netcfg as root
Click the Interfaces tab
Select the eth0 interface and Edit it
Change my IP number
Click Save

At this point the route command works again, and I can ping the other machine 
over the network.  However, my IP address has NOT YET CHANGED.  So something 
about the act of changing the IP address in netcfg has fixed the problem, 
even though the change has not yet taken effect.

I still can't make other changes in netcfg; for instance, if
I try to change my host domain (which is incorrect), I get
the following error:

SIOCDELRT: No such process
SIOCADDRT: No such device

Using ppp does not make the problem come back, but rebooting the computer 
(and thus finalizing the change to my IP address) does.

Ideas?  --Ben