We study the critical O(3) model using the numerical conformal bootstrap. In particular, we use a recently developed cutting-surface algorithm to efficiently map out the allowed space of CFT data from correlators involving the leading O(3) singlet s, vector \phi, and rank-2 symmetric tensor t. We determine their scaling dimensions to be (\Delta_{s}, \Delta_{\phi}, \Delta_{t}) = (0.518942(51), 1.59489(59), 1.20954(23)), and also bound various OPE coefficients. We additionally introduce a new "tip-finding" algorithm to compute an upper bound on the leading rank-4 symmetric tensor t_4, which we find to be relevant with \Delta_{t_4} < 2.99056. The conformal bootstrap thus provides a numerical proof that systems described by the critical O(3) model, such as classical Heisenberg ferromagnets at the Curie transition, are unstable to cubic anisotropy.
