  We demonstrate coherent control of both the rotational and hyperfine state of
ultracold, chemically stable $^{87}$Rb$^{133}$Cs molecules with external
microwave fields. We create a sample of ~2000 molecules in the lowest hyperfine
level of the rovibronic ground state N = 0. We measure the transition
frequencies to 8 different hyperfine levels of the N = 1 state at two magnetic
fields ~23 G apart. We determine accurate values of rotational and hyperfine
coupling constants that agree well with previous calculations. We observe Rabi
oscillations on each transition, allowing complete population transfer to a
selected hyperfine level of N = 1. Subsequent application of a second microwave
pulse allows transfer of molecules back to a different hyperfine level of N =
0.
