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We present the technique used to correct the geometric distortion in the Solar and Heliospheric Observatory/Solar Ultraviolet Measurement of Estimated Radiation (SOHO/SUMER) ultraviolet spectrograph images. Electric and magnetic fields within the SUMER crossed delay line detectors and optical misalignment within the SUMER spectrograph cause focal plane image distortion, which leads to errors in wavelength and position measurements if not corrected. A high degree of precision is required in geometric correction in order to achieve the required accuracy in solar ultraviolet line position measurements. We characterize the distortion using a combination of laboratory and in situ measurements to compute tables of horizontal and vertical pixel shift vectors and correct the distorted frame by interpolation to within 0.11 pixel rms error in wavelength pixel and 0.25 pixel rms error in spatial pixel. The correction allows measurements of line-of-sight solar ion velocities with 1 km/s accuracy.