The United States does not certify runways or approach systems to CAT IIIA or IIIB anymore. All CAT III runways are certified to fail operational capabilities, unless there is a problem with rollout guidance. Only a very few runways here have the transmissometers necessary to allow operations below RVR 600 feet, hence you'll see that as the published minimum in most cases. As Marker Inbound says, you must use the higher of either the runway minimums or your company minimums. The latter should reflect your aircraft capabilities, i.e., fail-passive or fail-operational.
There is still some misleading guidance floating around about fail-passive systems being limited to CAT IIIA. In fact, they have been limited to RVR 600 for several years now, which is less than CAT IIIA minimums. However, again, your company minimums must again be respected if they are greater than the published minimums. Keep an eye out for little notes on the chart that say things like "Localizer unusable for rollout". That can make a rather big difference!