The Geneva group has reported two Saturn-mass planets orbiting HD 83443 (K0 V) with periods of 2.98 and 29.8 days. The two planets have raised interest in their dynamics because of the possible 10:1 orbital resonance and the strong gravitational interactions. We report precise Doppler measurements of HD 83443 obtained with the Keck/HIRES and the Anglo-Australian Telescope (AAT) UCLES spectrometers. These measurements strongly confirm the inner planet with a period of 2.985 days, with orbital parameters in very good agreement with those of the Geneva group. However, these Doppler measurements show no evidence of the outer planet, at thresholds of one-fourth (3 m s-1) of the reported velocity amplitude of 13.8 m s-1. Thus, the existence of the outer planet is in question. Indeed, the current Doppler measurements reveal no evidence of any second planet with a period less than a year. Based on observations obtained at the Anglo-Australian Telescope, Siding Spring, Australia, and on observations obtained at the W. M. Keck Observatory, which is operated jointly by the University of California and the California Institute of Technology. Keck time has been granted by both NASA and the University of California.