We have measured parity-violating asymmetries in elastic electron-proton scattering over the range of momentum transfers 0.12 < or =Q2 < or =1.0 GeV2. These asymmetries, arising from interference of the electromagnetic and neutral weak interactions, are sensitive to strange-quark contributions to the currents of the proton. The measurements were made at Jefferson Laboratory using a toroidal spectrometer to detect the recoiling protons from a liquid hydrogen target. The results indicate nonzero, Q2 dependent, strange-quark contributions and provide new information beyond that obtained in previous experiments.