  We present a demonstration of delensing the observed cosmic microwave
background (CMB) B-mode polarization anisotropy. This process of reducing the
gravitational-lensing generated B-mode component will become increasingly
important for improving searches for the B modes produced by primordial
gravitational waves. In this work, we delens B-mode maps constructed from
multi-frequency SPTpol observations of a 90 deg$^2$ patch of sky by subtracting
a B-mode template constructed from two inputs: SPTpol E-mode maps and a lensing
potential map estimated from the $\textit{Herschel}$ $500\,\mu m$ map of the
CIB. We find that our delensing procedure reduces the measured B-mode power
spectrum by 28% in the multipole range $300 < \ell < 2300$; this is shown to be
consistent with expectations from theory and simulations and to be robust
against systematics. The null hypothesis of no delensing is rejected at $6.9
\sigma$. Furthermore, we build and use a suite of realistic simulations to
study the general properties of the delensing process and find that the
delensing efficiency achieved in this work is limited primarily by the noise in
the lensing potential map. We demonstrate the importance of including realistic
experimental non-idealities in the delensing forecasts used to inform
instrument and survey-strategy planning of upcoming lower-noise experiments,
such as CMB-S4.
