The detection of faint magnetic fields from single-electron and nuclear spins at the atomic scale is a long-standing challenge in physics. While current mobile quantum sensors achieve single-electron spin sensitivity, atomic spatial resolution remains elusive for existing techniques. Here we fabricate a single-molecule quantum sensor at the apex of the metallic tip of a scanning tunnelling microscope by attaching Fe atoms and a PTCDA (3,4,9,10-perylenetetracarboxylic-dianhydride) molecule to the tip apex. We address the molecular spin by electron spin resonance and achieve ~100 neV resolution in energy. In a proof-of-principle experiment, we measure the magnetic and electric dipole fields emanating from a single Fe atom and an Ag dimer on an Ag(111) surface with sub-angstrom spatial resolution. Our method enables atomic-scale quantum sensing experiments of electric and magnetic fields on conducting surfaces and may find applications in the sensing of spin-labelled biomolecules and of spin textures in quantum materials.