  We present an existence and stability theory for gravity-capillary solitary
waves with constant vorticity on the surface of a body of water of finite
depth. Exploiting a rotational version of the classical variational principle,
we prove the existence of a minimiser of the wave energy $\mathcal H$ subject
to the constraint $\mathcal I=2\mu$, where $\mathcal I$ is the wave momentum
and $0< \mu \ll 1$. Since $\mathcal H$ and $\mathcal I$ are both conserved
quantities a standard argument asserts the stability of the set $D_\mu$ of
minimisers: solutions starting near $D_\mu$ remain close to $D_\mu$ in a
suitably defined energy space over their interval of existence.
  In the applied mathematics literature solitary water waves of the present
kind are described by solutions of a Korteweg-deVries equation (for strong
surface tension) or a nonlinear Schr\"{o}dinger equation (for weak surface
tension). We show that the waves detected by our variational method converge
(after an appropriate rescaling) to solutions of the appropriate model equation
as $\mu \downarrow 0$
