  We show numerically that the `deconfined' quantum critical point between the
N\'eel antiferromagnet and the columnar valence-bond-solid, for a square
lattice of spin-1/2s, has an emergent $SO(5)$ symmetry. This symmetry allows
the N\'eel vector and the valence-bond-solid order parameter to be rotated into
each other. It is a remarkable 2+1-dimensional analogue of the $SO(4)=
[SU(2)\times SU(2)]/Z_2$ symmetry that appears in the scaling limit for the
spin-1/2 Heisenberg chain. The emergent $SO(5)$ is strong evidence that the
phase transition in the 2+1D system is truly continuous, despite the violations
of finite-size scaling observed previously in this problem. It also implies
surprising relations between correlation functions at the transition. The
symmetry enhancement is expected to apply generally to the critical
two-component Abelian Higgs model (non-compact $CP^1$ model). The result
indicates that in three dimensions there is an $SO(5)$-symmetric conformal
field theory which has no relevant singlet operators, so is radically different
to conventional Wilson-Fisher-type conformal field theories.
