We present measurements of polarization lensing using the 150 GHz maps which include all data taken by the BICEP2 & Keck Array CMB polarization experiments up to and including the 2014 observing season (BK14). Despite their modest angular resolution ($sim 0.5^circ$), the excellent sensitivity ($sim 3mu$K-arcmin) of these maps makes it possible to directly reconstruct the lensing potential using only information at larger angular scales ($ellleq 700$). From the auto-spectrum of the reconstructed potential we measure an amplitude of the spectrum to be $A^{phiphi}_{rm L}=1.15pm 0.36$ (Planck $Lambda$CDM prediction corresponds to $A^{phiphi}_{rm L}=1$), and reject the no-lensing hypothesis at 5.8$sigma$, which is the highest significance achieved to date using an EB lensing estimator. Taking the cross-spectrum of the reconstructed potential with the Planck 2015 lensing map yields $A^{phiphi}_{rm L}=1.13pm 0.20$. These direct measurements of $A^{phiphi}_{rm L}$ are consistent with the $Lambda$CDM cosmology, and with that derived from the previously reported BK14 B-mode auto-spectrum ($A^{rm BB}_{rm L}=1.20pm 0.17$). We perform a series of null tests and consistency checks to show that these results are robust against systematics and are insensitive to analysis choices. These results unambiguously demonstrate that the B-modes previously reported by BICEP / Keck at intermediate angular scales ($150lesssimelllesssim 350$) are dominated by gravitational lensing. The good agreement between the lensing amplitudes obtained from the lensing reconstruction and B-mode spectrum starts to place constraints on any alternative cosmological sources of B-modes at these angular scales.
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