We demonstrate coherent microwave control of the rotational, hyperfine and Zeeman states of ultracold CaF molecules, and the magnetic trapping of these molecules in a single, selectable quantum state. We trap about 5 X 10³ molecules for almost 2s at a temperature of 70(8) μK and a density of 1.2 X 10⁵ cm⁻³. We measure the state-specific loss rate due to collisions with background helium.