In this work, we present a complete analysis of the quantization of the classical Brans-Dicke theory using the method of affine quantization in the Hamiltonian description of the theory. The affine quantization method is based on the symmetry of the phase space of the system, in this case the (positive) half-plane, which is identified with the affine group. We consider a Friedmann-Lemaître-Robertson-Walker type spacetime, and since the scale factor is always positive, the affine method seems to be more suited than the canonical quantization for our quantum cosmology. We find the wave function of the Brans-Dicke universe and its energy spectrum. A smooth bounce is expected at the semiclassical level in the quantum phase-space portrait. We also address the problem of equivalence between the Jordan and Einstein frames.