  We discuss the relationship between exact solvability of the Schr\"{o}dinger
equation with a position-dependent mass and the ordering ambiguity in the
Hamiltonian operator within the frame of supersymmetric quantum mechanics. The
one-dimensional Schr\"{o}dinger equation, derived from the general form of the
effective mass Hamiltonian, is solved exactly for a system with exponentially
changing mass in the presence of a potential with similar behaviour, and the
corresponding supersymmetric partner Hamiltonians are related to the
effective-mass Hamiltonians proposed in the literature.
