  We report the discovery of X-ray eclipses in the recently discovered
accreting millisecond X-ray pulsar Swift J1749.4-2807. This is the first
detection of X-ray eclipses in a system of this type and should enable a
precise neutron star mass measurement once the companion star is identified and
studied. We present a combined pulse and eclipse timing solution that enables
tight constraints on the orbital parameters and inclination and shows that the
companion mass is in the range 0.6-0.8 M_sun for a likely range of neutron star
masses, and that it is larger than a main sequence star of the same mass. We
observed two individual eclipse egresses and a single ingress. Our timing model
shows that the eclipse features are symmetric about the time of 90 deg
longitude from the ascending node, as expected. Our eclipse timing solution
gives an eclipse duration (from the mid-points of ingress to egress) of 2172
+/- 13 s. This represents 6.85% of the 8.82 hr orbital period. This system also
presents a potential measurement of "Shapiro" delay due to General Relativity;
through this technique alone, we set an upper limit to the companion mass of
2.2 M_sun.
