[Abridged] We present two-dimensional line-of-sight stellar kinematics of the lens galaxy in the Einstein Cross, obtained with the GEMINI 8m telescope, using the GMOS integral-field spectrograph. The velocity map shows regular rotation up to ~100 km/s around the minor axis of the bulge, consistent with axisymmetry. The velocity dispersion map shows a weak gradient increasing towards a central (R<1) value of sigma_0=170+/-9 km/s. We deproject the observed surface brightness from HST imaging to obtain a realistic luminosity density of the lens galaxy, which in turn is used to build axisymmetric dynamical models that fit the observed kinematic maps. We also construct a gravitational lens model that accurately fits the positions and relative fluxes of the four quasar images. We find that the resulting luminous and total mass distribution are nearly identical around the Einstein radius R_E = 0.89, with a slope that is close to isothermal, but which becomes shallower towards the center if indeed mass follows light. The dynamical model fits to the observed kinematic maps result in a total mass-to-light ratio (M/L)_dyn=3.7+/-0.5 M_sun/L_sun,I (in the I-band). This is consistent with the Einstein mass M_E = 1.54 x 10^10 M_sun divided by the (projected) luminosity within R_E, which yields a total mass-to-light ratio of (M/L)_E=3.4 M_sun/L_sun,I, with an error of at most a few per cent. We estimate from stellar populations model fits to colors of the lens galaxy a stellar mass-to-light ratio (M/L)_* from 2.8 to 4.1 M_sun/L_sun,I. Although a constant dark matter fraction of 20 per cent is not excluded, dark matter may play no significant role in the bulge of this ~L* early-type spiral galaxy.
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