We present new ideas for evolving black holes through a computational grid without excision, which enable accurate and stable evolutions of binary black hole systems with the accurate determination of gravitational waveforms directly from the wave zone region. Rather than excising the black hole interiors, our approach follows the "puncture" treatment of black holes, but utilizing a new gauge condition which allows the black holes to move successfully through the computational domain. We apply these techniques to an inspiraling binary, modeling the radiation generated during the final plunge and ringdown. We demonstrate convergence of the waveforms and good conservation of mass-energy, with just over 3% of the system's mass converted to gravitational radiation.