  (Abridged) We present the results of variable star searches of the M31 dwarf
spheroidal companions Andromeda I and Andromeda III using the Hubble Space
Telescope. A total of 100 variable stars were found in Andromeda I, while 56
were found in Andromeda III. One variable found in Andromeda I and another in
Andromeda III may be Population II Cepheids. In addition to this variable in
Andromeda III, another four variables are anomalous Cepheids. We found 72
fundamental mode RR Lyrae stars and 26 first-overtone mode RR Lyrae stars in
Andromdea I giving mean periods of 0.575 day and 0.388 day, respectively. One
likely RR Lyrae was not classified. For Andromeda III, 39 RR Lyrae stars are
pulsating in the fundamental mode with a mean period of 0.657 day and 12 are in
the first-overtone mode with a mean period of 0.402 day. While Andromeda I has
a mean period that is consistent with its metallicity, the mean period of the
fundamental mode RR Lyrae in Andromeda III is larger than expected. In
Andromeda I, we found two RR Lyrae stars which are noticeably fainter than the
horizontal branch. We discuss the possibility that these stars are associated
with the recently discovered stellar stream in the halo of M31. We find
<V_RR>=25.14+/-0.04 mag for Andromeda I resulting in a distance of 765+/-25
kpc. For Andromeda III, <V_RR>=25.01+/-0.04 mag giving a distance of 740+/-20
kpc. The relation between specific frequency of anomalous Cepheids and mean
metallicity of the galaxy in the M31 dwarf spheroidal galaxies follows the same
relation as the Galactic dwarf spheroidal galaxies.
