A quantization of unimodular gravity is described, which results in a quantum
effective action which is also unimodular, ie a function of a metric with fixed
determinant. A consequence is that contributions to the energy momentum tensor
of the form of the metric times a spacetime constant, whether classical or
quantum, are not sources of curvature in the equations of motion derived from
the quantum effective action. This solves the first cosmological constant
problem, which is suppressing the enormous contributions to the cosmological
constant coming from quantum corrections. We discuss several forms of uniodular
gravity and put two of them, including one proposed by Henneaux and Teitelboim,
in constrained Hamiltonian form. The path integral is constructed from the
latter. Furthermore, the second cosmological constant problem, which is why the
measured value is so small, is also addressed by this theory. We argue that a
mechanism first proposed by Ng and van Dam for suppressing the cosmological
constant by quantum effects obtains at the semiclassical level.Comment: 22 pages, no figure