The narrow-line Seyfert 1 galaxy IRAS 13224−3809 has been observed with XMM-Newton for 500 ks. The source is rapidly variable on time-scales down to a few 100 s. The spectrum shows strong broad Fe − K and L emission features which are interpreted as arising from reflection from the inner parts of an accretion disc around a rapidly spinning black hole. Assuming a power law emissivity for the reflected flux and that the innermost radius corresponds to the innermost stable circular orbit, the black hole spin is measured to be 0.989 with a statistical precision better than 1 per cent. Systematic uncertainties are discussed. A soft X-ray lag of 100 s confirms this scenario. The bulk of the power-law continuum source is located at a radius of 2-3 gravitational radii.