  We describe a simple model for calculating the zero-temperature superfluid
density of Zn-doped YBa_2Cu_3O_{7-\delta} as a function of the fraction x of
in-plane Cu atoms which are replaced by Zn. The basis of the calculation is a
``Swiss cheese'' picture of a single CuO_2 layer, in which a substitutional Zn
impurity creates a normal region of area $\pi\xi_{ab}^2$ around it as
originally suggested by Nachumi et al. Here $\xi_{ab}$ is the zero-temperature
in-plane coherence length at x = 0. We use this picture to calculate the
variation of the in-plane superfluid density with x at temperature T = 0, using
both a numerical approach and an analytical approximation. For $\delta = 0.37$,
if we use the value $\xi_{ab}$ = 18.3 angstrom, we find that the in-plane
superfluid decreases with increasing x and vanishes near $x_c = 0.01$ in the
analytical approximation, and near $x_c = 0.014$ in the numerical approach.
$x_c$ is quite sensitive to $\xi_{ab}$, whose value is not widely agreed upon.
The model also predicts a peak in the real part of the conductivity,
Re$\sigma_e(\omega, x)$, at concentrations $x \sim x_c$, and low frequencies,
and a variation of critical current density with x of the form $J_c(x) \propto
n_{S,e}(x)^{7/4}$ near percolation, where $n_{S,e}(x)$ is the in-plane
superfluid density.
