bnt,
It is clear that the gateway address is 10.0.0.138, and the PC IP address is 10.0.0.1.
Unusual, but hey, there's nothing wrong with it.
DHCP is already set for the PC - but it appears that it isn't working for DNS server settings, for whatever reason, as it is assigning the gateway address as the DNS server.
There's nothing peculiar about the fact that the Mac is working. It obviously has valid DNS settings.
The "destination can not be reached" is the response you expect with the default gateway being configured as the DNS server - but you are right to say that the PC must therefore be talking to the gateway.
Restarting the router and then running "ipconfig /release" and "ipconfig /renew" might help to reset the DNS config.
I have had this problem before - and resorted to fixed IP addresses and manual DNS configuration. It is a LOT more reliable! And technically a little bit more secure!
SD