We report on the discovery and the timing analysis of the first eclipsing accretion-powered millisecond X-ray pulsar (AMXP): SWIFT J1749.4-2807. The neutron star rotates at a frequency of ~517.9 Hz and is in a binary system with an orbital period of 8.8 hr and a projected semimajor axis of ~1.90 lt-s. Assuming a neutron star between 0.8 and 2.2 M sun and using the mass function of the system and the eclipse half-angle, we constrain the mass of the companion and the inclination of the system to be in the ~0.46-0.81 M sun and ~ 74fdg4-77fdg3 range, respectively. To date, this is the tightest constraint on the orbital inclination of any AMXP. As in other AMXPs, the pulse profile shows harmonic content up to the third overtone. However, this is the first AMXP to show a first overtone with rms amplitudes between ~6% and ~23%, which is the strongest ever seen and which can be more than two times stronger than the fundamental. The fact that SWIFT J1749.4-2807 is an eclipsing system that shows uncommonly strong harmonic content suggests that it might be the best source to date to set constraints on neutron star properties including compactness and geometry.