We develop a theory for the rapid search of specific sites on DNA, via a mechanism in which a nonspecifically-bound protein can switch between two conformations. In the "inactive" conformation, the bound protein has favorable, nonspecific interactions with the DNA, but cannot be recognized by the target site. In the "active" conformation, the protein is recognized by the target site but has a very rugged energy surface elsewhere on the DNA. The rate constant for protein binding to the specific site is calculated by an approach in which the protein, after reaching the DNA surface via 3D diffusion, searches for the target site via 1D diffusion while being allowed to escape to the bulk solution. Mindful of the pitfalls of many previous approximate treatments, we validate our approach against a rigorous solution of the problem when the protein has a fixed conformation. In the 1D diffusion toward the specific site, a conformationally switchable protein predominantly adopts the inactive conformation due to the favorable nonspecific interactions with the DNA, thus maximizing the 1D diffusion constant and minimizing the chance of escape to the bulk solution. Once at the target site, a transition to the active conformation allows the protein to be captured. This induced-switch mechanism provides robust speedup of protein-DNA binding rates, and appears to be adopted by many transcription factors and DNA-modifying enzymes.