The paper solves by Riemann's method the one-dimensional problem of the expansion of a gas cloud into a vacuum. The cloud is initially at rest, and is homogeneous except for a boundary layer of arbitrary thickness. Motion starts in the non-homogeneous layer, and spreads into the cloud. The solution uses Riemann's variables |$r=\frac{1}{2}(3c+u),\,s=\frac{1}{2}(3c+u)$|, where c is the local velocity of sound, u the gas velocity. In the first stage of the motion there are three moving boundaries. In |$x\gt {x}_{1}(t)$|, there is a vacuum; in |${x}_{2}(t)\lt x\lt {x}_{1}(t)$|, r and s are both variable; in |${x}_{3}(t)\lt x\lt {x}_{2}(t)$|, r is constant; in |$x\lt {x}_{3}(t)$|, the gas is homogeneous and at rest. The boundaries |$x={x}_{1}(t)\,\text{and}\,x={x}_{2}(t)$| move to the right; and, as x2 increases more rapidly than x1, the two boundaries coincide at the end of the first stage of the motion. The boundary |$x={x}_{3}(t)$| moves into the cloud with the local velocity of sound. In the second stage of the motion, the quantity r is constant every Where. The cloud still advances into the vacuum, and the motion spreads as before into the cloud. This state is similar to one discovered by Burgers. The difficulty regarding the discontinuity in the initial state of Burgers' solution is shown to be due to the fact that his solution is a limiting case of the present solution when the thickness of the initial layer of non-homogeneity is made to vanish.