  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 scaling limit of such a dynamics,
derived through a scaling of the jump rate. Informally, we expect that, in the
limit, only jumps of ``infinite length'' will survive, i.e., we expect to
arrive at a Glauber dynamics in continuum (a birth-and-death process in
$\mathbb{R}^d$). We prove that, in the low activity-high temperature regime,
the generators of the Kawasaki dynamics converge to the generator of a Glauber
dynamics. The convergence is on the set of exponential functions, in the
$L^2(\mu)$-norm. Furthermore, additionally assuming that the potential of pair
interaction is positive, we prove the weak convergence of the
finite-dimensional distributions of the processes.
