The violation of mirror symmetry in the weak force provides a powerful tool to study the internal structure of the proton. Experimental results have been obtained that address the role of strange quarks in generating nuclear magnetism. The measurement reported here provides an unambiguous constraint on strange quark contributions to the proton's magnetic moment through the electron-proton weak interaction. We also report evidence for the existence of a parity-violating electromagnetic effect known as the anapole moment of the proton. The proton's anapole moment is not yet well understood theoretically, but it could have important implications for precision weak interaction studies in atomic systems such as cesium.