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We report the first measurement of analyzing powers and spin correlation parameters using a storage ring with both beam and internal target polarized. Spin observables were measured for elastic scattering of 45 and 198 MeV protons from polarized [sup 3]He nuclei in a new laser-pumped internal gas target at the Indiana University Cyclotron Facility Cooler Ring. Scattered protons and recoil [sup 3]He nuclei were detected in coincidence with large acceptance plastic scintillators and silicon detectors. The internal-target technique demonstrated in this experiment has broad applicability to the measurement of spin-dependent scattering in nuclear and particle physics.