We consider a cellular system with amplify-and-forward (AF) nonconcurrent two-way relaying (ncTWR), where a base station serves a transmit-only user and a receive-only user. Most of the state-of-the-art transceiver designs for AF multiple-input multiple-output (MIMO) ncTWR optimize a system-wide objective function subject to the transmit power constraints. Transceiver designs that incorporate quality-of-service (QoS) constraints are not well investigated in ncTWR literature. In this paper, we design a MIMO AF transceiver that maximizes weighted sum-rate (WSR) while guaranteeing the QoS constraints that are cast as per-stream rate required by two ncTWR users. The WSR maximization is a nonconvex problem due to its nonconvex objective. We solve this problem by separately approximating the objective at low and high signal-to-noise ratios (SNRs), with each approximation cast as a geometric program. With extensive numerical evaluations, we first demonstrate the improved performance of the proposed transceiver over existing designs without QoS constraints. We later investigate the effect of QoS constraints on the system WSR. © 1967-2012 IEEE.