  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 $\xi \approx 350,000$
lattice spacings -- more than three orders of magnitude larger than any
previous study. We resolve a conundrum concerning the applicability of
asymptotic-scaling formulae to experimentally- and numerically-determined
correlation lengths, and arrive at a very precise determination of the
low-energy observables. Our results have direct implications for the
zero-temperature behavior of spin-1/2 ladders.
