The self-assembly of amphiphilic molecules in solution is a ubiquitous process in both natural and synthetic systems. The ability to effectively control the structure and properties of these systems is essential for tuning the quality of their functionality, yet the underlying mechanisms governing the transition from molecules to assemblies have not been fully resolved. Here we describe how amphiphilic self-assembly can be preceded by liquid-liquid phase separation. The assembly of a model block co-polymer system into vesicular structures was probed through a combination of liquid-phase electron microscopy, self-consistent field computations and Gibbs free energy calculations. This analysis shows the formation of polymer-rich liquid droplets that act as a precursor in the bottom-up formation of spherical micelles, which then evolve into vesicles. The liquid-liquid phase separation plays a role in determining the resulting vesicles' structural properties, such as their size and membrane thickness, and the onset of kinetic traps during self-assembly.