  We prove exponential decay of transverse correlations in the Spin O(N) model
for arbitrary (non-zero) values of the external magnetic field and arbitrary
spin dimension N > 1. Our result is new when N > 3, in which case no Lee-Yang
theorem is available, it is an alternative to Lee-Yang when N = 2, 3, and also
holds for a wide class of multi-component spin systems with continuous
symmetry. The key ingredients are a representation of the model as a system of
coloured random paths, a `colour-switch' lemma, and a sampling procedure which
allows us to bound from above the `typical' length of the open paths.
