We present a measurement of the $B$-mode polarization power spectrum (the $BB$ spectrum) from 100 $mathrm{deg}^2$ of sky observed with SPTpol, a polarization-sensitive receiver currently installed on the South Pole Telescope. The observations used in this work were taken during 2012 and early 2013 and include data in spectral bands centered at 95 and 150 GHz. We report the $BB$ spectrum in five bins in multipole space, spanning the range $300 le ell le 2300$, and for three spectral combinations: 95 GHz $times$ 95 GHz, 95 GHz $times$ 150 GHz, and 150 GHz $times$ 150 GHz. We subtract small ($< 0.5 sigma$ in units of statistical uncertainty) biases from these spectra and account for the uncertainty in those biases. The resulting power spectra are inconsistent with zero power but consistent with predictions for the $BB$ spectrum arising from the gravitational lensing of $E$-mode polarization. If we assume no other source of $BB$ power besides lensed $B$ modes, we determine a preference for lensed $B$ modes of $4.9 sigma$. After marginalizing over tensor power and foregrounds, namely polarized emission from galactic dust and extragalactic sources, this significance is $4.3 sigma$. Fitting for a single parameter, $A_mathrm{lens}$, that multiplies the predicted lensed $B$-mode spectrum, and marginalizing over tensor power and foregrounds, we find $A_mathrm{lens} = 1.08 pm 0.26$, indicating that our measured spectra are consistent with the signal expected from gravitational lensing. The data presented here provide the best measurement to date of the $B$-mode power spectrum on these angular scales.
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