  We consider the general spin-1 SU(2) invariant Heisenberg model with a
two-body interaction. A random loop model is introduced and relations to
quantum spin systems is proved. Using this relation it is shown that for
dimensions 3 and above N\'eel order occurs for a large range of values of the
relative strength of the bilinear ($-J_1$) and biquadratic ($-J_2$) interaction
terms. The proof uses the method of reflection positivity and infrared bounds.
Links between spin correlations and loop correlations are proved.
