  We consider a natural variant of Berezin-Toeplitz quantization of compact
K\"{a}hler manifolds, in the presence of a Hamiltonian circle action lifting to
the quantizing line bundle. Assuming that the moment map is positive, we study
the diagonal asymptotics of the associated Szeg\"{o} and Toeplitz operators,
and specifically their relation to the moment map and to the geometry of a
certain symplectic quotient. When the underlying action is trivial and the
moment map is taken to be identically equal to one, this scheme coincides with
the usual Berezin-Toeplitz quantization. This continues previous work on
near-diagonal scaling asymptotics of equivariant Szeg\"{o} kernels in the
presence of Hamiltonian torus actions.
