Results are presented for the first measurement of the double-polarization helicity asymmetry E for the η photoproduction reaction γp→ηp. Data were obtained using the FROzen Spin Target (FROST) with the CLAS spectrometer in Hall B at Jefferson Lab, covering a range of center-of-mass energy W from threshold to 2.15 GeV and a large range in center-of-mass polar angle. As an initial application of these data, the results have been incorporated into the J\”ulich model to examine the case for the existence of a narrow N∗ resonance between 1.66 and 1.70 GeV. The addition of these data to the world database results in marked changes in the predictions for the E observable using that model. Further comparison with several theoretical approaches indicates these data will significantly enhance our understanding of nucleon resonances.