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 conformal field theory 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(phi), Delta(s), Delta(t)) = (0.518942(51), 1.59489(59), 1.20954(23)), and also bound various operator product expansion 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(t4) < 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.