Quantum metrology uses quantum entanglement-correlations in the properties of microscopic systems-to improve the statistical precision of physical measurements(1). When measuring a signal, such as the phase shift of a light beam or an atomic state, a prominent limitation to achievable precision arises from the noise associated with the counting of uncorrelated probe particles. This noise, commonly referred to as shot noise or projection noise, gives rise to the standard quantum limit (SQL) to phase resolution. However, it can be mitigated down to the fundamental Heisenberg limit by entangling the probe particles. Despite considerable experimental progress in a variety of physical systems, a question that persists is whether these methods can achieve performance levels that compare favourably with optimized conventional (non-entangled) systems. Here we demonstrate an approach that achieves unprecedented levels of metrological improvement using half a million Rb-87 atoms in their 'clock' states. The ensemble is 20.1 +/- 0.3 decibels (100-fold) spin-squeezed via an optical-cavity-based measurement. We directly resolve small microwave-induced rotations 18.5 +/- 0.3 decibels (70-fold) beyond the SQL. The single-shot phase resolution of 147 microradians achieved by the apparatus is better than that achieved by the best engineered cold atom sensors despite lower atom numbers(2,3). We infer entanglement of more than 680 +/- 35 particles in the atomic ensemble. Applications include atomic clocks(4), inertial sensors(5), and fundamental physics experiments such as tests of general relativity(6) or searches for electron electric dipole moment(7). To this end, we demonstrate an atomic clock measurement with a quantum enhancement of 10.5 +/- 0.3 decibels (11-fold), limited by the phase noise of our microwave source.