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We have determined lithium abundances for a sample of 79 halo subgiants. The subgiant candidates were identified using uvby photometry from several catalogs of metal-poor stars. The basic data were high-resolution, low-noise coude spectra in the 6700 A spectral region. Abundances of iron and calcium, derived from one Ca I and several Fe I lines in our spectra, provided a metallicity discriminant for the stars in our sample. The subgiants with temperatures between 5500 and 4900 K show a steady decline of lithium abundance with advancing subgiant position (and decreasing temperature). The observed trend is in qualitative agreement with recent theoretical models of lithium dilution in metal-poor stars, especially if main-sequence diffusion is included. The initial lithium abundances in metal-poor stars may have been slightly larger than that exhibited by stars near the main-sequence turnoff. For stars with temperatures below 4900 K, the models predict no further dilution, but observed lithium abundances continue to decline with decreasing temperature, indicating further lithium destruction on the giant branch of metal-poor stars. In all postdilution subgiants, the observed lithium abundances show more scatter than do stars at the main-sequence turnoff, suggesting variations in the main-sequence lithium destruction below the observable surface layers.