The enormous size and cost of current state-of-the-art accelerators based on conventional radio-frequency technology has spawned great interest in the development of new acceleration concepts that are more compact and economical. Micro-fabricated dielectric laser accelerators (DLAs) are an attractive approach, because such dielectric microstructures can support accelerating fields one to two orders of magnitude higher than can radio-frequency cavity-based accelerators. DLAs use commercial lasers as a power source, which are smaller and less expensive than the radio-frequency klystrons that power today's accelerators. In addition, DLAs are fabricated via low-cost, lithographic techniques that can be used for mass production. However, despite several DLA structures having been proposed recently(1-4), no successful demonstration of acceleration in these structures has so far been shown. Here we report high-gradient (beyond 250 MeV m(-1)) acceleration of electrons in a DLA. Relativistic (60-MeV) electrons are energy-modulated over 563 +/- 104 optical periods of a fused silica grating structure, powered by a 800-nm-wavelength mode-locked Ti:sapphire laser. The observed results are in agreement with analytical models and electrodynamic simulations. By comparison, conventional modern linear accelerators operate at gradients of 10-30 MeV m(-1), and the first linear radio-frequency cavity accelerator was ten radio-frequency periods (one metre) long with a gradient of approximately 1.6 MeV m(-1) (ref. 5). Our results set the stage for the development of future multi-staged DLA devices composed of integrated on-chip systems. This would enable compact table-top accelerators on the MeV-GeV (10(6)-10(9) eV) scale for security scanners and medical therapy, university-scale X-ray light sources for biological and materials research, and portable medical imaging devices, and would substantially reduce the size and cost of a future collider on the multi-TeV (10(12) eV) scale.