Large Language Models (LLMs) are increasingly deployed as Internet/Web services (LLM-as-a-Service) with strict latency Service-Level Objectives (SLOs) under tight GPU memory budgets. Mixture-of-Experts (MoE) models improve quality and throughput via sparse expert activation, but serving them efficiently is challenging because expert weights dominate memory footprint and incur costly host--device transfers when offloaded. Moreover, MoE serving exhibits a phase disparity: the prefill phase tends to activate experts densely across many tokens, while the decode phase activates only a few experts per step. A uniform expert loading/caching policy across phases leads to either peak-memory blowup (prefill) or tail-latency inflation (decode). We present DuoServe-MoE, a QoS-oriented MoE serving system that decouples prefill and decode and applies phase-specialized expert scheduling. For prefill, DuoServe-MoE uses a two-stream CUDA pipeline to overlap expert prefetching with non-MoE computation, reducing expert residency time and peak GPU memory. For decode, it employs a lightweight layer-level predictor trained offline from activation traces to prefetch only likely experts without model changes. Experiments on representative MoE LLMs show that DuoServe-MoE improves TTFT by up to $5.34\times$ and end-to-end latency by up to $7.55\times$ over representative baselines, while maintaining low runtime GPU memory usage under resource-constrained deployment.