Compact, table-top sized accelerators are key to improving access to high-quality beams for use in industry, medicine and academic research. Among laser-based accelerating schemes, the inverse free-electron laser (IFEL) enjoys unique advantages. By using an undulator magnetic field in combination with a laser, GeV m(-1) gradients may be sustained over metre-scale distances using laser intensities several orders of magnitude less than those used in laser wake-field accelerators. Here we show for the first time the capture and high-gradient acceleration of monoenergetic electron beams from a helical IFEL. Using a modest intensity (~10(13) W cm(-2)) laser pulse and strongly tapered 0.5 m long undulator, we demonstrate >100 MV m(-1) accelerating gradient, >50 MeV energy gain and excellent output beam quality. Our results pave the way towards compact, tunable GeV IFEL accelerators for applications such as driving soft X-ray free-electron lasers and producing γ-rays by inverse Compton scattering.