  We derive the one loop mixing matrix for anomalous dimensions in N=4 Super
Yang-Mills. We show that this matrix can be identified with the Hamiltonian of
an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz
to find a recipe for computing anomalous dimensions for a wide range of
operators. We give exact results for BMN operators with two impurities and
results up to and including first order 1/J corrections for BMN operators with
many impurities. We then use a result of Reshetikhin's to find the exact
one-loop anomalous dimension for an SO(6) singlet in the limit of large bare
dimension. We also show that this last anomalous dimension is proportional to
the square root of the string level in the weak coupling limit.
