The dynamics of phospholipids in unilamellar vesicles (ULVs) is of interest in biology, medical, and food sciences, since these molecules are widely used as biocompatible agents and a mimic of cell membrane systems. We have investigated the nanoscopic dynamics of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) phospholipid in ULVs as a function of temperature using elastic and quasielastic neutron scattering (QENS). The dependence of the signal on the scattering momentum transfer, which is a critical advantage of neutron scattering techniques, allows the detailed analysis of the lipid motions that cannot be carried out by other means. In agreement with a differential scanning calorimetry measurement, a sharp rise in the elastic scattering intensity below ca. 296 K indicates a phase transition from the high-temperature fluid phase to the low-temperature solid gel phase. The microscopic lipid dynamics exhibits qualitative differences between the solid gel phase (in a measurement at 280 K) and the fluid phase (in a measurement at a physiological temperature of 310 K). The analysis of the data demonstrates the presence of two types of distinct motions: the entire lipid molecule motion within a monolayer, also known as lateral diffusion, and the relatively faster internal motion of the DMPC molecule. The lateral diffusion of the entire lipid molecule is Fickian in character, whereas the internal lipid motions are of localized character, which is consistent with the structure of the vesicles. The lateral motion slows down by an order of magnitude in the solid gel phase, whereas for the internal motion not only the time scale but also the character of the motion changes upon the phase transition. In the solid gel phase, the lipids are more ordered and undergo uniaxial rotational motion. However, in the fluid phase, the hydrogen atoms of the lipid tails undergo confined translation diffusion rather than uniaxial rotational diffusion. The translational, but spatially localized, diffusion of the hydrogen atoms of the lipid tails is a manifestation of the flexibility of the chains acquired in the fluid phase. Because of this flexibility, both the local diffusivity and the confinement volume for the hydrogen atoms increase in the linear fashion from near the lipid's polar headgroup to the end of its hydrophobic tail. Our results present a quantitative and detailed picture of the effect of the gel-fluid phase transition on the nanoscopic lipid dynamics in ULVs. The data analysis approach developed here has a potential for probing the dynamic response of lipids to the presence of additional cell membrane components.