X-ray phase-contrast imaging has recently led to a revolution in resolving power and tissue contrast in biomedical imaging, microscopy and materials science. The necessary high spatial coherence is currently provided by either large-scale synchrotron facilities with limited beamtime access or by microfocus X-ray tubes with rather limited flux. X-rays radiated by relativistic electrons driven by well-controlled high-power lasers offer a promising route to a proliferation of this powerful imaging technology. A laser-driven plasma wave accelerates and wiggles electrons, giving rise to a brilliant keV X-ray emission. This so-called betatron radiation is emitted in a collimated beam with excellent spatial coherence and remarkable spectral stability. Here we present a phase-contrast microtomogram of a biological sample using betatron X-rays. Comprehensive source characterization enables the reconstruction of absolute electron densities. Our results suggest that laser-based X-ray technology offers the potential for filling the large performance gap between synchrotron- and current X-ray tube-based sources.