Nuclear magnetic resonance (NMR) spectroscopy is a useful tool in understanding molecular composition and dynamics, but simulating NMR spectra of large molecules becomes intractable on classical computers as the spin correlations in these systems can grow exponentially with molecule size. In contrast, quantum computers are well suited to simulate NMR spectra of molecules, particularly zero- to ultralow field (ZULF) NMR where the spin-spin interactions in the molecules dominate. In this work, we demonstrate the first quantum simulation of an NMR spectrum, specifically that of the methyl group of acetonitrile in ZULF, using a trapped ion quantum computer. The simulation involves state-of-the-art “QFAST” circuit synthesis algorithm that produces short circuits, with the circuit sampling rate considerably reduced by employing a compressed sensing technique. This work lays the foundation for simulation of NMR experiments on near-term quantum hardware.