Complete flat surfaces in Rn are studied under the condition that the normal connection is flat and the length of the mean curvature vector is constant. It is shown that such a surface must be the product of two curves of constant geodesic curvature. In this paper we prove the following theorem. Theorem. Let M be a complete C' flat surface in Ri . Suppose that the normal connection of M is flat and the length of the mean curvature vector is constant. Then there exist curves of constant geodesic curvature C1 in Ir and C2 in Rn-r (1 < r < n 1) such that M is congruent to the Riemannian product of C1 and C2 . A surface is called flat if the Gaussian curvature is zero at every point. Let M be a surface in R . We denote the standard inner product and the covariant differentiation of Rn by ( , ) and D respectively. For tangent vector fields X, Y, and a normal vector field 4 of M, we write DX Y = DXY + B(X, Y) and DxX = -A<X + Dx, where DxY (resp. -AX) is the tangential component of DX Y (resp. DxX ), and B(X, Y) (resp. DX4) is the normal component of DXY (resp. Dx ). Let {ei} (i = 1, 2) be a local orthonormal frame field of the tangent bundle TM of M and {ea,, } (a = 3, ... , n) be a local orthonormal frame field of the normal bundle TIM of M. We define Wi AB(X) = (DXeA, eB) (A, B = 1, ..., n) for X in TM. We say that a point p in M is umbilical with respect to a normal vector 4 at p if A< is proportional to the identity transformation of the tangent space T M of M at p. If K denotes the Gaussian curvature of M, the Gauss equation is given by (1) K = (B(e, , e,), B(e2, e2)) (B(el , e2), B(e, , e2)). Received by the editors July 28, 1989 and, in revised form, October 19, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C42; Secondary 53A07.