Mold cannot grow on stone. It needs a food source, usually cellulosic (plant-based) in nature, as well as moisture.
If you have mold growing on marble, it is likely growing on biofilm, a buildup of biological material on the stone. Vacuum the marble thoroughly with a unit with a HEPA-rated filter to remove spores and prevent the mold from spreading. You can use a fungicide like Benefect, which will not harm marble, to disinfect. Then remove the biofilm by cleaning the stone, and the mold will not be able to grow.
Any "staining" (the filamentous roots of the mold) left behind can be removed with a marble-safe poultice.