  A Kawasaki dynamics in continuum is a dynamics of an infinite system of
interacting particles in $\mathbb R^d$ which randomly hop over the space. In
this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs
measure $\mu$ as invariant measure. We study a diffusive limit of such a
dynamics, derived through a scaling of both the jump rate and time. Under weak
assumptions on the potential of pair interaction, $\phi$, (in particular,
admitting a singularity of $\phi$ at zero), we prove that, on a set of smooth
local functions, the generator of the scaled dynamics converges to the
generator of the gradient stochastic dynamics. If the set on which the
generators converge is a core for the diffusion generator, the latter result
implies the weak convergence of finite-dimensional distributions of the
corresponding equilibrium processes. In particular, if the potential $\phi$ is
from $C_{\mathrm b}^3(\mathbb R^d)$ and sufficiently quickly converges to zero
at infinity, we conclude the convergence of the processes from a result in
[Choi {\it et al.}, J. Math. Phys. 39 (1998) 6509--6536].
