  We present the results of a search for additional bodies in the GJ 1132
system through two methods: photometric transits and transit timing variations
of the known planet. We collected 21 transit observations of GJ 1132b with the
MEarth-South array since 2015. We obtained 100 near-continuous hours of
observations with the $Spitzer$ Space Telescope, including two transits of GJ
1132b and spanning 60\% of the orbital phase of the maximum period at which
bodies coplanar with GJ 1132b would pass in front of the star. We exclude
transits of additional Mars-sized bodies, such as a second planet or a moon,
with a confidence of 99.7\%. When we combine the mass estimate of the star
(obtained from its parallax and apparent $K_s$ band magnitude) with the stellar
density inferred from our high-cadence $Spitzer$ light curve (assuming zero
eccentricity), we measure the stellar radius of GJ 1132 to be
$0.2105^{+0.0102}_{-0.0085} R_\odot$, and we refine the radius measurement of
GJ 1132b to $1.130 \pm 0.056 R_\oplus$. Combined with HARPS RV measurements, we
determine the density of GJ 1132b to be $6.2 \pm 2.0$\ g cm$^{-3}$, with the
mass determination dominating this uncertainty. We refine the ephemeris of the
system and find no evidence for transit timing variations, which would be
expected if there was a second planet near an orbital resonance with GJ 1132b.
