  Adapting ideas of Daubechies and Klauder [J. Math. Phys. {\bf 26} (1985)
2239] we derive a rigorous continuum path-integral formula for the semigroup
generated by a spin Hamiltonian. More precisely, we use spin-coherent vectors
parametrized by complex numbers to relate the coherent representation of this
semigroup to a suitable Schr\"odinger semigroup on the Hilbert space $L^2(R^2)$
of Lebesgue square-integrable functions on the Euclidean plane $R^2$. The
path-integral formula emerges from the standard Feynman-Kac-It\^o formula for
the Schr\"odinger semigroup in the ultra-diffusive limit of the underlying
Brownian bridge on $R^2$. In a similar vein, a path-integral formula can be
constructed for the coherent representation of the unitary time evolution
generated by the spin Hamiltonian.
