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We examine models of the mass distribution for the first known case ofgravitational lensing. Several new sets of constraints are used, based onrecent observations. We remodel the VLBI observations of the radio jets in thetwo images of Q0957+561, showing that the previously derived positions anduncertainties were incorrect. We use as additional constraints the candidatelensed pairs of galaxies identified recently with the Hubble Space Telescope.We explore a wider range of lens models than before, and find that the Hubbleconstant is not tightly constrained once elliptical lens models are considered.We also discuss the systematic uncertainties caused by the cluster containingthe lens galaxy. We conclude that additional observations of the candidatelensed galaxies as well as direct measurements of the cluster mass profile areneeded before an accurate value of the Hubble constant can be derived from thislens.Comment: 28 pages, Latex + 4 figures, accepted for ApJ. Final version, with  minor changes. Model parameters in Table 4 correcte

A Reassessment of the Data and Models of the Gravitational Lens Q0957+561