The Cryogenic Dark Matter Search (CDMS) employs low-temperature Ge and Si detectors to search for weakly interacting massive particles (WIMPs) via their elastic-scattering interactions with nuclei while discriminating against interactions of background particles. For recoil energies above 10 keV, events due to background photons are rejected with >99.9% efficiency, and surface events are rejected with >95% efficiency. The estimate of the background due to neutrons is based primarily on the observation of multiple-scatter events that should all be neutrons. Data selection is determined primarily by examining calibration data and vetoed events. Resulting efficiencies should be accurate to ∼10%. Results of CDMS data from 1998 and 1999 with a relaxed fiducial-volume cut (resulting in 15.8 kg days exposure on Ge) are consistent with an earlier analysis with a more restrictive fiducial-volume cut. Twenty-three WIMP candidate events are observed, but these events are consistent with a background from neutrons in all ways tested. Resulting limits on the spin-independent WIMP-nucleon elastic-scattering cross section exclude unexplored parameter space for WIMPs with masses between 10–70 GeV/c2. These limits border, but do not exclude, parameter space allowed by supersymmetry models and accelerator constraints. Results are compatible with some regions reported as allowed at 3σ by the annual-modulation measurement of the DAMA Collaboration. However, under the assumptions of standard WIMP interactions and a standard halo, the results are incompatible with the DAMA most likely value at >99.9% confidence level (C.L.), and are incompatible with the model-independent annual-modulation signal of DAMA at 99.99% C.L. in the asymptotic limit.