We compare the angular expansion velocities, determined with VLBI, with the linear expansion velocities measured from optical spectra for supernova 1993J in the galaxy M81, over the period from 7 d to ~9 yr after shock breakout. We estimate the distance to SN 1993J using the Expanding Shock Front Method (ESM). We find the best distance estimate is obtained by fitting the angular velocity of a point halfway between the contact surface and outer shock front to the maximum observed hydrogen gas velocity. We obtain a direct, geometric, distance estimate for M81 of D=3.96+-0.05+-0.29 Mpc with statistical and systematic error contributions, respectively, corresponding to a total standard error of $+-0.29 Mpc. The upper limit of 4.25 Mpc corresponds to the hydrogen gas with the highest observed velocity reaching no farther out than the contact surface a few days after shock breakout. The lower limit of 3.67 Mpc corresponds to this hydrogen gas reaching as far out as the forward shock for the whole period, which would mean that Rayleigh-Taylor fingers have grown to the forward shock already a few days after shock breakout. Our distance estimate is 9+-13 % larger than that of 3.63+-0.34 Mpc from the HST Key Project, which is near our lower limit but within the errors.
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