The detection and characterization of an Earth-like planet orbiting a nearby star requires a telescope with an extraordinarily large contrast at small angular separations. At visible wavelengths, an Earth-like planet would be 1 × 10-10 times fainter than the star at angular separations of typically 0.1 arcsecond or less. There are several proposed space telescope systems that could, in principle, achieve this. Here we report a laboratory experiment that reaches these limits. We have suppressed the diffracted and scattered light near a star-like source to a level of 6 × 10-10 times the peak intensity in individual coronagraph images. In a series of such images, together with simple image processing, we have effectively reduced this to a residual noise level of about 0.1 × 10-10. This demonstrates that a coronagraphic telescope in space could detect and spectroscopically characterize nearby exoplanetary systems, with the sensitivity to image an 'Earth-twin' orbiting a nearby star.