We present striking results from Rossi X-Ray Timing Explorer (RXTE) observations of the 21 hr low-mass X-ray binary 4U 1624-49, showing five complex dips in unprecedented detail. For the first time, dipping is detected up to 15 keV. Prominent flares are also observed in the light curves, limited to energies above ~8 keV. Spectra selected by intensity during dip episodes can be well fitted with a two-component model consisting of a pointlike blackbody from the neutron star and progressive covering of an extended Comptonized region, presumably an accretion disk corona (ADC), corrected for photons scattered into and out of the X-ray beam by a interstellar dust halo. We find that the outer regions of the absorber are highly ionized and that electron scattering is totally responsible for the X-ray attenuation during shallow dipping. The timescales of dip ingress and egress indicate that the envelope of material absorbing the ADC has smaller angular size than the ADC itself and that the ADC is likely limited to a height-to-radius ratio of 10% rather than being spherical in extent. In addition, we have analyzed ~4.5 yr of RXTE All-Sky Monitor (ASM) coverage to derive the first accurate orbital ephemeris for 4U 1624-49, with phase zero (the time of dip centers) well described by the relation 2,450,088.63918(69) + N×0. 869907(12) (JD).