Memory safety has long been a critical challenge in software engineering, particularly for legacy systems written in memory-unsafe languages such as C and C++. Rust, one of the youngest modern programming languages, offers built-in memory-safety guarantees that make it a strong candidate for secure systems development. Consequently, transpiling C/C++ code into memory-safe Rust code has become a growing area of research. However, manual transpilation is often time-consuming and error-prone. Additionally, rule-based automated approaches are not as flexible or cost-effective as methods enabled by state-of-the-art AI models, techniques, and methods, such as those that deploy Large Language Models (LLMs), for example, Generative Pretrained Transformers (GPT). In this paper, we propose a Retrieval-Augmented Generation (RAG)-assisted framework that integrates an LLM with a Small Language Model (SLM) to perform C/C++-to-Rust transpilation with a focus on enhancing memory safety. The framework deploys a segmentation strategy that processes C/C++ code in balanced blocks, guiding the LLM with retrieved context from Rust documentation and compiler error references. Our experiments using three OpenAI models (GPT-4o, GPT-4-Turbo, and o3-Mini) demonstrate that the RAG-enhanced pipeline generally improves both code correctness and security for C-to-Rust code transpilation. Several Coreutils programs achieve complete elimination of Raw Pointer Dereferences (RPDs) and Unsafe Type Casts (UTCs) in the final Rust output, indicating the potential of LLM-based transpilation for advancing automated software modernization and repair, as well as memory-safe code generation.