The Seyfert 1 galaxy Zw 229-015 is among the brightest active galaxies being monitored by the Kepler mission. In order to determine the black hole mass in Zw 229-015 from H beta reverberation mapping, we have carried out nightly observations with the Kast Spectrograph at the Lick 3 m telescope during the dark runs from 2010 June through December, obtaining 54 spectroscopic observations in total. We have also obtained nightly V-band imaging with the Katzman Automatic Imaging Telescope at Lick Observatory and with the 0.9 m telescope at the Brigham Young University West Mountain Observatory over the same period. We detect strong variability in the source, which exhibited more than a factor of two change in broad H beta flux. From cross-correlation measurements, we find that the H beta light curve has a rest-frame lag of 3.86(-0.90)(+0.69) days with respect to the V-band continuum variations. We also measure reverberation lags for H alpha and H gamma and find an upper limit to the H delta lag. Combining the H beta lag measurement with a broad H beta width of sigma(line) = 1590 +/- 47 km s(-1) measured from the rms variability spectrum, we obtain a virial estimate of M-BH = 1.00(-0.24)(+0.19) x 10(7) M-circle dot for the black hole in Zw 229-015. As a Kepler target, Zw 229-015 will eventually have one of the highest-quality optical light curves ever measured for any active galaxy, and the black hole mass determined from reverberation mapping will serve as a benchmark for testing relationships between black hole mass and continuum variability characteristics in active galactic nuclei.