[en] 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 deg² 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 Herschel 500 μm map of the cosmic infrared background. We find that our delensing procedure reduces the measured B-mode power spectrum by $28$% in the multipole range $300<<!--\; <\; -->\mathrm{\ell <!--\; ???\; -->}<<!--\; <\; -->2300;$ this is shown to be consistent with expectations from simulations and to be robust against systematics. The null hypothesis of no delensing is rejected at $6.9\mathrm{\sigma <!--\; ??\; -->}$. 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.