The interworking of the LTE system and WLAN technologies has drawn much attention lately, due to the growing demands for various multimedia services and large data traffic in hotspot areas. Existing research studies have mostly investigated the coupling architectures for these two wireless communication standards at the network layer. However, in the current architectures, many important coordination functions and joint optimizations cannot be accomplished efficiently. To tackle this problem, a new CBS solution is proposed, which integrates different RATs at layer 2 in the true sense of convergence. We design a unified protocol stack that includes all the original functions of both LTE and WLAN systems. Then we propose a convergence architecture, the RMC sublayer, for joint management of these two RATs. The proposed CBS solution can support seamless offloading through soft handover, guaranteed QoS, forwarding management by a single IP address, and customized bandwidth aggregation service. Finally, our simulation and initial experiment results demonstrate the feasibility and efficiency of the CBS solution in future mobile converged networks.