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Analysis of several recent ROSAT HRI observations of the gravitationallylensed system Q0957+561 has led to the detection at the 3sigma level of thecluster lens containing the primary galaxy G1. The total mass was estimated byapplying the equation of hydrostatic equilibrium to the detected hotintracluster gas for a range of cluster core radii, cluster sizes and fordifferent values of the Hubble constant. X-ray estimates of the lensing clustermass provide a means to determine the cluster contribution to the deflection ofrays originating from the quasar Q0957+561. The present mass estimates wereused to evaluate the convergence parameter kappa, the ratio of the localsurface mass density of the cluster to the critical surface mass density forlensing. The convergence parameter, kappa, calculated in the vicinity of thelensed images, was found to range between 0.07 and 0.21, depending on theassumed cluster core radius and cluster extent. This range of uncertainty inkappa does not include possible systematic errors arising from the estimationof the cluster temperature through the use of the clusterluminosity-temperature relation and the assumption of spherical symmetry of thecluster gas. Applying this range of values of kappa to the lensing model ofGrogin & Narayan (1996) for Q0957+561 but not accounting for uncertainties inthat model yields a range of values for the Hubble constant:67<H_0<82 km s^-1Mpc^-1, for a time delay of 1.1 years.Comment: Accepted for publication in ApJ, 25 pages, 9 figure

X‐Ray Detection of the Primary Lens Galaxy Cluster of the Gravitational Lens System Q0957+561