  We consider the time-independent Wigner functions of phase-space quantum
mechanics (a.k.a. deformation quantization) for a Morse potential. First, we
find them by solving the $\ast$-eigenvalue equations, using a method that can
be applied to potentials that are polynomial in an exponential. A Mellin
transform converts the $\ast$-eigenvalue equations to difference equations, and
factorized solutions are found directly for all values of the parameters. The
symbols of both diagonal and off-diagonal density operator elements in the
energy basis are found this way. The Wigner transforms of the density matrices
built from the known wave functions are then shown to confirm the solutions.
