  We consider the Wiener sausage for a Brownian motion with a constant drift up
to time $t$ associated with a closed ball. In the two or more dimensional
cases, we obtain the explicit form of the expected volume of the Wiener
sausage. The result says that it can be represented by the sum of the mean
volumes of the multi-dimensional Wiener sausages without a drift. In addition,
we show that the leading term of the expected volume of the Wiener sausage is
written as $\kappa t(1+o[1])$ for large $t$ and an explicit form gives the
constant $\kappa$. The expression is of a complicated form, but it converges to
the known constant as the drift tends to $0$.
