We report the first precision measurement of the proton electric to magnetic form factor ratio from spin-dependent elastic scattering of longitudinally polarized electrons from a polarized hydrogen internal gas target. The measurement was performed at the MIT-Bates South Hall Ring over a range of four-momentum transfer squared Q2 from 0.15 to 0.65 (GeV/c)(2). Significantly improved results on the proton electric and magnetic form factors are obtained in combination with existing cross-section data on elastic electron-proton scattering in the same Q2 region.