Hi jswain,
your situation is entirely different. You are running avast inside the VM box, as opposed to the original poster, who ran avast on the host.
It might help, if you would post some info about the versions and settings you are using. We have already tried that here, with MAC OS 10.7.3, VirtualBox (v4.1.12) and there are no apparent problems with network access in both NAT and Bridge network modes.
Lukas