We characterize those non-negative, measurable functions ψ on [0, 1] and positive, continuous functions ω1 and ω2 on ⁺ for which the generalized Hardy-Cesàro operator defines a bounded operator Uψ: L ¹(ω1) → L ¹(ω2) This generalizes a result of Xiao [7] to weighted spaces. Furthermore, we extend Uψ to a bounded operator on M(ω1) with range in L ¹(ω2) δ0, where M(ω1) is the weighted space of locally finite, complex Borel measures on ⁺. Finally, we show that the zero operator is the only weakly compact generalized Hardy-Cesàro operator from L ¹(ω1) to L ¹(ω2).