The spatial instability problem in a slowly diverging rectangular duct is investigated. The mean flow for the present problem is three-dimensional and has been obtained asymptotically using lubrication theory. Using a WKBJ expansion for the disturbance quantities, the zeroth- and first-order equations are derived. The zeroth-order problem corresponds to a locally parallel flow approximation and the first-order problem yields the non-parallel-flow correction to the eigenvalues obtained from the former through the use of a solvability condition. The solution of these equations is discussed and the results used to determine the effect of the variation in duct geometry on the neutral curves.