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Mass spectrometry is a powerful and widely used tool for identifying molecular structures due to its sensitivity and ability to profile complex samples. However, translating spectra into full molecular structures is a difficult, under-defined inverse problem. Overcoming this problem is crucial for enabling biological insight, discovering new metabolites, and advancing chemical research across multiple fields. To this end, we develop MSFlow, a two-stage encoder-decoder flow-matching generative model that achieves state-of-the-art performance on the structure elucidation task for small molecules. In the first stage, we adopt a formula-restricted transformer model for encoding mass spectra into a continuous and chemically informative embedding space, while in the second stage, we train a decoder flow matching model to reconstruct molecules from latent embeddings of mass spectra. We present ablation studies demonstrating the importance of using information-preserving molecular descriptors for encoding mass spectra and motivate the use of our discrete flow-based decoder. Our rigorous evaluation demonstrates that MSFlow can accurately translate up to 45 percent of molecular mass spectra into their corresponding molecular representations - an improvement of up to fourteen-fold over the current state-of-the-art. A trained version of MSFlow is made publicly available on GitHub for non-commercial users.