  If $X$ is a smooth curve such that the minimal degree of its plane models is
not too small compared with its genus, then $X$ has been known to be a double
cover of another smooth curve $Y$ under some mild condition on the genera.
However there are no results yet for the minimal degrees of plane models of
double covers except some special cases. In this paper, we give upper and lower
bounds for the minimal degree of plane models of the double cover $X$ in terms
of the gonality of the base curve $Y$ and the genera of $X$ and $Y$. In
particular, the upper bound equals to the lower bound in case $Y$ is
hyperelliptic. We give an example of a double cover which has plane models of
degree equal to the lower bound.
