This paper presents a new algorithm for evolving orbiting black-hole binaries without requiring excision or a corotating shift. The algorithm is based on a novel technique to handle the singular puncture conformal factor. The system, based on the BSSN formulation of Einstein's equations, is non-singular at the start of the evolution and remains nonsingular and stable with a good choice of gauge. The technique is tested by fully evolving orbiting black-hole binaries from near the Innermost Stable Circular Orbit (ISCO) regime. The results show fourth-order convergence of waveforms and compute the radiated gravitational energy and angular momentum from the plunge. These results agree well with those predicted by the Lazarus approach. The main goal of numerical relativity is to compute accurate gravitational waveforms from astrophysical simulations of merging black-hole binaries. The expectation of strong gravitational wave emission from black hole mergers, along with recent astrophysical observations, makes these systems important for study. The numerical relativity community has made significant efforts to solve the binary-black-hole problem, leading to the development of new formulations and numerical techniques. The calculation of gravitational radiation from plunging black-hole binaries was pioneered using the Lazarus approach, which bridges numerical relativity and perturbative techniques. More recently, progress has been made in evolving orbiting binary-black-hole spacetimes using stable full 3D numerical relativity codes with corotating gauge conditions and singularity excision. The paper presents a novel technique for evolving orbiting black holes based on puncture data. This technique does not require a corotating shift or singularity excision. The method is based on the BSSN formulation and uses a new variable to regularize the system near the puncture. The evolution equations are derived and shown to be explicitly finite on the initial slice. The method is tested with simulations showing fourth-order convergence of waveforms and accurate computation of radiated energy and angular momentum. The paper also discusses the use of a 'multiple transition' Fisheye transformation to handle the wide range of scales in black-hole binary systems. The results show good agreement with the Lazarus approach and demonstrate the effectiveness of the new technique in simulating black-hole binaries. The paper concludes with future plans to study more interesting astrophysical scenarios and to use more robust wave extraction methods.This paper presents a new algorithm for evolving orbiting black-hole binaries without requiring excision or a corotating shift. The algorithm is based on a novel technique to handle the singular puncture conformal factor. The system, based on the BSSN formulation of Einstein's equations, is non-singular at the start of the evolution and remains nonsingular and stable with a good choice of gauge. The technique is tested by fully evolving orbiting black-hole binaries from near the Innermost Stable Circular Orbit (ISCO) regime. The results show fourth-order convergence of waveforms and compute the radiated gravitational energy and angular momentum from the plunge. These results agree well with those predicted by the Lazarus approach. The main goal of numerical relativity is to compute accurate gravitational waveforms from astrophysical simulations of merging black-hole binaries. The expectation of strong gravitational wave emission from black hole mergers, along with recent astrophysical observations, makes these systems important for study. The numerical relativity community has made significant efforts to solve the binary-black-hole problem, leading to the development of new formulations and numerical techniques. The calculation of gravitational radiation from plunging black-hole binaries was pioneered using the Lazarus approach, which bridges numerical relativity and perturbative techniques. More recently, progress has been made in evolving orbiting binary-black-hole spacetimes using stable full 3D numerical relativity codes with corotating gauge conditions and singularity excision. The paper presents a novel technique for evolving orbiting black holes based on puncture data. This technique does not require a corotating shift or singularity excision. The method is based on the BSSN formulation and uses a new variable to regularize the system near the puncture. The evolution equations are derived and shown to be explicitly finite on the initial slice. The method is tested with simulations showing fourth-order convergence of waveforms and accurate computation of radiated energy and angular momentum. The paper also discusses the use of a 'multiple transition' Fisheye transformation to handle the wide range of scales in black-hole binary systems. The results show good agreement with the Lazarus approach and demonstrate the effectiveness of the new technique in simulating black-hole binaries. The paper concludes with future plans to study more interesting astrophysical scenarios and to use more robust wave extraction methods.