A cubic spline method is developed for the linear two-point boundary value problem \(u'' + b(x)u' + c(x)u = f\), \(x \in [0,1]\); \(\lambda_ su'(s) + \mu_ su(s) = \nu_ s\), \(s = 0,1\). The method is a discrete version of the \(H^ 1\)-Galerkin method, yields a linear algebraic system with bandwidth 5 and has a rate of convergence \(O(h^{4-i})\) in the \(W^ i_ p\)-norm for \(i = 0,1,2\); \(1 \leq p \leq \infty\). An orthogonal spline collocation with Hermite cubics is briefly discussed.