We present the results of deep spectroscopy for the central region of the cluster lens SDSS J1004+4112 with the Subaru telescope. A secure detection of an emission line of the faint blue stellar object (component E) near the center of the brightest cluster galaxy (G1) confirms that it is the central fifth image of the lensed quasar system. In addition, we measure the stellar velocity dispersion of G1 to be sigma_* = 352+-13 km/s. We combine these results to obtain constraints on the mass M_BH of the putative black hole (BH) at the center of the inactive galaxy G1, and hence on the M_BH-sigma_* relation at the lens redshift z_l=0.68. From detailed mass modeling, we place an upper limit on the black hole mass, M_BH < 2.1x10^{10}M_sun at 1-sigma level (<3.1x10^{10}M_sun at 3-sigma), which is consistent with black hole masses expected from the local and redshift-evolved M_BH-sigma_* relations, M_BH~10^{9}-10^{10}M_sun.
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