We present a first cosmological analysis of a refined cluster catalog from the Red-Sequence Cluster Survey (RCS). The input cluster sample is derived from the deepest 72.07 deg2 of the RCS images, which probe to the highest redshift and lowest mass limits. The catalog contains 956 clusters over 0.35<z<0.95, limited by cluster richness and richness error. The calibration of the survey images has been extensively cross-checked against publicly available Sloan Digital Sky Survey imaging, and the cluster redshifts and richness that result from this well-calibrated subset of data are robust. We analyze the cluster sample via a general self-calibration technique. We fit simultaneously for the matter density, Ωm, the normalization of the power spectrum, σ8, and four parameters describing the calibration of cluster richness to mass, its evolution with redshift, and scatter in the mass-richness relation. The principal goal of this general analysis is to establish the consistency (or lack thereof) between the fitted parameters (both cosmological and cluster mass observables) and available results on both from independent measures. From an unconstrained analysis, the derived values of Ωm and σ8 are 0.31+0.11-0.10 and 0.67+0.18-0.13, respectively. An analysis including Gaussian priors on the slope and zero point of the mass-richness relation gives very similar results: 0.30+0.12-0.11 and 0.70+0.27-0.15. Both analyses are in acceptable agreement with the current literature. The derived parameters describing the mass-richness relation in the unconstrained fit are also eminently reasonable and in good agreement with existing follow-up data on both the RCS-1 and other cluster samples. Our results directly demonstrate that future surveys (optical and otherwise), with much larger samples of clusters, can give constraints competitive with other probes of cosmology. Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.