Each successive generation of X-ray machines has opened up new frontiers in science, such as the first radiographs and the determination of the structure of DNA. State-of-the-art X-ray sources can now produce coherent high-brightness Xrays of greater than kiloelectronvolt energy and promise a new revolution in imaging complex systems on nanometre and femtosecond scales. Despite the demand, only a few dedicated synchrotron facilities exist worldwide, in part because of the size and cost of conventional (accelerator) technology(1). Here we demonstrate the use of a new generation of laser-driven plasma accelerators(2), which accelerate high-charge electron beams to high energy in short distances(3-5), to produce directional, spatially coherent, intrinsically ultrafast beams of hard X-rays. This reduces the size of the synchrotron source from the tens of metres to the centimetre scale, simultaneously accelerating and wiggling the electron beam. The resulting X-ray source is 1,000 times brighter than previously reported plasma wigglers(6,7) and thus has the potential to facilitate a myriad of uses across the whole spectrum of light-source applications.