The correlation length of the square-lattice spin-1/2 Heisenberg antiferromagnet is studied in the low-temperature (asymptotic-scaling) regime. Our novel approach combines a very efficient loop cluster algorithm―operating directly in the Euclidean time continuum―with finite-size scaling. This enables us to probe correlation lengths up to ξ≈350000 lattice spacings, more than 3 orders of magnitude larger than in any previous study. We resolve a conundrum concerning the applicability of asymptotic-scaling formulas to experimentally and numerically determined correlation lengths. Our results have direct implications for the zero-temperature behavior of spin-1/2 ladders.