Atomic hydrogen Lyman alpha radiation (121.6 nm) has been measured in emission from the atmosphere of Ganymede with the Galileo ultraviolet spectrometer. An exospheric model with the following parameters has been fit to the observational data: atomic hydrogen density directly above the surface (radius 2634 km) equal to 1.5¿104 atoms cm-3, scale height 2634 km, exospheric temperature 450 K. A model calculation of the photodissociation of water vapor from surface ice at 146 K is used to obtain the photodissociation rate necessary to supply the hydrogen atoms that are escaping from the exosphere of Ganymede. The calculated escape flux of atomic hydrogen is 7¿108 atoms/cm2 sec. Two alternate but speculative sources of the atomic hydrogen escaping from Ganymede are photodesorption of water ice by ultraviolet photons in the wavelength range 120.5-186.0 nm and sputtering of water ice by Jupiter's magnetospheric ion plasma.¿ 1997 American Geophysical Union