In contrast to classical empty space, the quantum vacuum fundamentally alters the properties of embedded particles. This paradigm shift allows one to explain the discovery of the celebrated Lamb shift in the spectrum of the hydrogen atom. Here, we engineer a synthetic vacuum, building on the unique properties of ultracold atomic gas mixtures, offering the ability to switch between empty space and quantum vacuum. Using high-precision spectroscopy, we observe the phononic Lamb shift, an intriguing many-body effect originally conjectured in the context of solid-state physics.We find good agreement with theoretical predictions based on the Fröhlich model. Our observations establish this experimental platform as a new tool for precision benchmarking of open theoretical challenges, especially in the regime of strong coupling between the particles and the quantum vacuum.