Precipitation is expected in Titan's atmosphere, yet it has not been directly observed, and the geographical regions where rain occurs are unknown. Here we present near-infrared spectra from the Very Large Telescope and W. M. Keck Observatories that reveal an enhancement of opacity in Titan's troposphere on the morning side of the leading hemisphere. Retrieved extinction profiles are consistent with condensed methane in clouds at an altitude near 30 kilometers and concomitant methane drizzle below. The moisture encompasses the equatorial region over Titan's brightest continent, Xanadu. Diurnal temperature gradients that cause variations in methane relative humidity, winds, and topography may each be a contributing factor to the condensation mechanism. The clouds and precipitation are optically thin at 2.0 micrometers, and models of "subvisible" clouds suggest that the droplets are 0.1 millimeter or larger.