A relaying scheme that uses both radio frequency (RF) and free space optical (FSO) links is investigated. In the proposed quantize and encode relaying (QER), a mobile source broadcasts the RF signal using quadrature amplitude modulation (QAM). The relay estimates and quantizes the log-likelihood ratio (LLR) of each received bit in the symbol and transmits the corresponding information to the destination through a high-speed FSO link or through a hybrid FSO/RF link. Relaying of multiple bits through high-speed FSO links improves RF spectrum utilization over systems where both the source and the relay transmit using RF signals. Exploiting the high available unlicensed bandwidth of the FSO links, the relay can even encode the LLRs using a channel code. Multiple encoded bits can then be mapped to FSO or hybrid FSO/RF symbols. The performance of the proposed system is analyzed, and it is shown that optimal symbol mapping can provide improvements of several decibels over conventional FSO transmissions that employ ON and OFF for binary ones and zeros, respectively. The system is found to achieve full diversity in fading FSO and RF channels.