  Results are presented from high-precision computations of the orbital
evolution and emitted gravitational waves for a stellar-mass object spiraling
into a massive black hole in a slowly shrinking, circular, equatorial orbit.
The focus of these computations is inspiral near the innermost stable circular
orbit (isco)---more particularly, on orbits for which the angular velocity
Omega is 0.03 < Omega/Omega_{isco} < 1. The computations are based on the
Teukolsky-Sasaki-Nakamura formalism, and the results are tabulated in a set of
functions that are of order unity and represent relativistic corrections to
low-orbital-velocity formulas. These tables can form a foundation for future
design studies for the LISA space-based gravitational-wave mission. A first
survey of applications to LISA is presented: Signal to noise ratios S/N are
computed and graphed as functions of the time-evolving gravitational-wave
frequency for representative values of the hole's mass M and spin a and the
inspiraling object's mass \mu, with the distance to Earth chosen to be r_o = 1
Gpc. These S/N's show a very strong dependence on the black-hole spin, as well
as on M and \mu. A comparison with predicted event rates shows strong promise
for detecting these waves, but not beyond about 1Gpc if the inspiraling object
is a white dwarf or neutron star. This argues for a modest lowering of LISA's
noise floor. A brief discussion is given of the prospects for extracting
information from the observed waves
