From inelastic neutron-scattering experiments and all atom molecular dynamics simulations we present evidence for a low-energy dynamical mode in the fluid phase of a 1,2-dimyristoyl-sn-glycero-3-phoshatidylcholine (DMPC) bilayer immersed in a 5% water/ethanol solution. In addition to the well-known phonon that shows a liquidlike dispersion with energies up to 4.5 meV, we observe an additional mode at smaller energies of 0.8 meV, which shows little or no dispersion. Both modes show transverse properties and might be related to molecular motion perpendicular to the bilayer.