This work studies the joint transceiver design for a full-duplex (FD) cloud radio access network (C- RAN) with simultaneous wireless information and power transfer (SWIPT). In the considered network, a number of FD remote radio heads (RRHs) receive information from uplink users (UUs), while transmitting both information and energy to a set of half-duplex (HD) downlink users (DUs) with power splitting receivers. Based on the particular problem structure, a block coordinate descent (BCD) method is proposed to minimize the total transmission power subject to both uplink-downlink quality of service (QoS) constraints and energy harvesting (EH) constraints. Although the problem has complicated constraints coupling a set of transceivers, uplink transmit power levels, and receive power splitting ratios, we prove that the proposed BCD algorithm converges to a Karush-Kuhn- Tucker (KKT) solution. Simulation results validate the effectiveness of the proposed algorithm as compared with the traditional HD scheme.