In this paper we treat the problem of transferring mass at least cost from one line segment to another, when there is a continuous cost function c(x, y) giving the cost of transferring material from the point x on the first line segment to the point y on the second. The mass has to be arranged with uniform density on the second line segment after the transfer. This is a one-dimensional form of the well-known mass-transfer problem. It is an infinite-dimensional linear program. We discuss duality theory for this problem and give an algorithm which converges to an optimal solution.