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The correlation length of the square-lattice spin-1/2 Heisenbergantiferromagnet 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 anyprevious study. We resolve a conundrum concerning the applicability ofasymptotic-scaling formulae to experimentally- and numerically-determinedcorrelation lengths, and arrive at a very precise determination of thelow-energy observables. Our results have direct implications for thezero-temperature behavior of spin-1/2 ladders.Comment: 12 pages, RevTeX, plus two Postscript figures. Some minor  modifications for final submission to Physical Review Letters. (accepted by  PRL

Square-Lattice Heisenberg Antiferromagnet at Very Large Correlation Lengths