Remove affected plants promptly before the black resting spores are formed and contaminate the soil. In the case of garden mint it is also necessary to remove infected rhizomes. In an infected bed, try to locate any uninfected stems and carefully dig these out and move to another location in an attempt to start a new, healthy colony.
Heat treatment is a method used by commercial nurseries, but could be adapted for the home gardener. Wash rhizomes thoroughly in early autumn and immerse in hot water at 44ºC (111ºF) (no higher) for 10 minutes, then cool in cold water and plant. An accurate thermometer is required, because 44ºC (111ºF) is very near the lethal temperature for the plant, and it may be more profitable to spend the money on some new plants.
No fungicides are available to gardeners to treat rust on mint used for food.
For ornamental species, fungicides containing myclobutanil (Doff Systemic Fungus Control and other formulations), tebuconazole (Bayer Garden Multirose concentrate2) or triticonazole (Scotts Fungus Clear Ultra) may be used. They will be most effective at preventing new infections, but because they have systemic activity (are taken into the plant tissues) they may have some curative properties.
Fungicides for gardeners (Adobe Acrobat pdf document outlining fungicides available to gardeners).