This paper proposes a new per-class bandwidth constraint algorithm, namely multipath selection algorithm (MSA), for a DiffServ-aware traffic engineering (DiffServ-TE). The MSA is achieved in three steps. First, the source uses the MSA to find multiple label switch paths (LSPs) from the source to the destination for a class type (CT). Second, the source uses the available bandwidth of the CT on all the links along these found LSPs to allocate the initial traffic for these selected LSPs. Third, the source dynamically adjusts traffics for each LSPs based on their transmission time. The simulation results have shown that the proposed algorithm has better performance in the average transmission time, packet loss rate, throughput and available bandwidth variance of each link than those based on the existing approach.