> Cats excrete the pathogen in their feces for a number of weeks after contracting the disease, generally by eating an infected intermediate host that could include mammals (like rodents) or birds. Oocyst shedding usually starts from the third day after ingestion of infected intermediate hosts, and may continue for weeks. The oocysts are not infective when excreted. After about a day, the oocyst undergoes a process called sporulation and becomes potentially pathogenic.

I guess I was not quite correct -- it may be that it sticks around longer in the cat, but it is only excreted for a few weeks.

0: https://en.wikipedia.org/wiki/Toxoplasmosis#Transmission