  We investigate the nature of the long-period radial velocity variations in
Alpha Tau first reported over 20 years ago. We analyzed precise stellar radial
velocity measurements for Alpha Tau spanning over 30 years. An examination of
the Halpha and Ca II 8662 spectral lines, and Hipparcos photometry was also
done to help discern the nature of the long-period radial velocity variations.
Our radial velocity data show that the long-period, low amplitude radial
velocity variations are long-lived and coherent. Furthermore, Halpha equivalent
width measurements and Hipparcos photometry show no significant variations with
this period. Another investigation of this star established that there was no
variability in the spectral line shapes with the radial velocity period. An
orbital solution results in a period of P = 628.96 +/- 0.90 d, eccentricity, e
= 0.10 +/- 0.05, and a radial velocity amplitude, K = 142.1 +/- 7.2 m/s.
Evolutionary tracks yield a stellar mass of 1.13 +/- 0.11 M_sun, which
corresponds to a minimum companion mass of 6.47 +/- 0.53 M_Jup with an orbital
semi-major axis of a = 1.46 +/- 0.27 AU. After removing the orbital motion of
the companion, an additional period of ~ 520 d is found in the radial velocity
data, but only in some time spans. A similar period is found in the variations
in the equivalent width of Halpha and Ca II. Variations at one-third of this
period are also found in the spectral line bisector measurements. The 520 d
period is interpreted as the rotation modulation by stellar surface structure.
Its presence, however, may not be long-lived, and it only appears in epochs of
the radial velocity data separated by $\sim$ 10 years. This might be due to an
activity cycle. The data presented here provide further evidence of a planetary
companion to Alpha Tau, as well as activity-related radial velocity variations.
