  We report the discovery of the orbital period of the ultracompact low-mass
X-ray binary (LMXB) 4U 1543-624 using time-resolved optical photometry taken
with the 6.5-m Clay (Magellan II) telescope in Chile. The light curve in the
Sloan r' band clearly shows a periodic, sinusoidal modulation at 18.2+-0.1 min
with a fractional semiamplitude of 8%, which we identify as the binary period.
This is the second shortest orbital period among all the known LMXBs, and it
verifies the earlier suggestion of 4U 1543-624 as an ultracompact binary based
on X-ray spectroscopic properties. The sinusoidal shape of the optical
modulation suggests that it arises from X-ray heating of the mass donor in a
relatively low-inclination binary, although it could also be a superhump
oscillation in which case the orbital period is slightly shorter. If the donor
is a C-O white dwarf as previously suggested, its likely mass and radius are
around 0.03 M_sun and 0.03 R_sun, respectively. For conservative mass transfer
onto a neutron star and driven by gravitational radiation, this implies an
X-ray luminosity of 6.5X10^36 erg/s and a source distance of 7 kpc. We also
discuss optical photometry of another LMXB, the candidate ultracompact binary
4U 1822-000. We detected significant optical variability on a time scale of
about 90 min, but it is not yet clear whether this was due to a periodic
modulation.
