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The index structures represented by the learned indexes are crucial components of storage systems. However, their performance is restricted by the memory bandwidth/latency wall in conventional computer architectures. Processing-in-memory (PIM) technology is a promising solution by integrating processing units directly into memory devices. In this paper, we propose PIMLex, a well-designed learned index with PIM, to alleviate the memory-bound issue. PIMLex overcomes the capacity limitations of existing PIM hardware by employing a decoupled two-layer structure. This design simultaneously leverages the powerful data processing capabilities of PIM and the large capacity of conventional DRAM. Additionally, a PIM-friendly model structure is incorporated to minimize computational tasks that PIM struggles with. Combined with a hotness-aware replication mechanism that ensures load balancing across numerous PIM modules, PIMLex is able to deliver high performance across various workload patterns. We implement PIMLex on UPMEM, an available commercial PIM. PIMLex achieves 36.5× higher throughput than the PIM-based learned index baseline and 2.2× higher than the DRAM-based ALEX.