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Laser-pumped dye amplifiers are the most convenient way to amplify ultrashort light pulses. In this paper, we develop an analytical theory of a transverse-pumped dye amplifier and compute the steady-state distribution of the excited state population and the total stored energy. The equation for the traveling amplified pulse is then solved for a given distribution of excited state molecules. The efficiency of an amplification stage associated with the distortion of the temporal pulse shape is obtained as a function of the input and stored energy density, normalized to the saturation level. The theoretical results are then compared to measurements obtained from an experimental arrangement of a three-stage optically pumped dye laser amplifier, which amplifies subpicosecond pulses from a passively mode-locked CW dye laser, to produce pulses with a peak intensity of 3 GW while maintaining a 0.5 picosecond pulsewidth.