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 κt(1+o [1]) for large t and an explicit form gives the constant κ. The expression is of a complicated form, but it converges to the known constant as the drift tends to 0.2010 Mathematics Subject Classification. Primary 60J65; Secondary 44A10.