© 2016 The Author(s). There is urgent need to develop new acceleration techniques capable of exceeding gigaelectron-volt-per-metre (GeV m-1) gradients in order to enable future generations of both light sources and high-energy physics experiments. To address this need, short wavelength accelerators based on wakefields, where an intense relativistic electron beam radiates the demanded fields directly into the accelerator structure or medium, are currently under intense investigation. One such wakefield based accelerator, the dielectric wakefield accelerator, uses a dielectric lined-waveguide to support a wakefield used for acceleration. Here we show gradients of 1.347±0.020 GeV m-1using a dielectric wakefield accelerator of 15 cm length, with sub-millimetre transverse aperture, by measuring changes of the kinetic state of relativistic electron beams. We follow this measurement by demonstrating accelerating gradients of 320±17 MeV m-1. Both measurements improve on previous measurements by and order of magnitude and show promise for dielectric wakefield accelerators as sources of high-energy electrons.