The Texas Animal Health Department tells me:
In the U.S. and other parts of the world - the West Nile virus doesn't seem to be a problem in domestic animals. Many different animal species can get a low level infection but easily fight off the virus without showing any symptoms.
Of course, there are always exceptions. Horses and wild birds can get very sick. Domestic birds, according to the health department, have no real problems with the West Nile virus.