We carried out a series of molecular dynamics simulations in order to examine the evaporative cooling of a nanoscale droplet of a Lennard-Jones liquid. After thermally equilibrating a droplet at a temperature T-ini/T-t similar or equal to 1.2 (T-t is the triple-point temperature), we started the evaporation into vacuum by removing vaporized particles and monitoring the change in droplet size and the temperature inside. As free evaporation proceeds, the droplet reaches a deep supercooled liquid state of T/T-t similar or equal to 0.7. The temperature was found to be uniform in spite of the fast evaporative cooling on the surface. The time evolution of the evaporating droplet properties was satisfactorily explained with a simple one-dimensional phase-change model. After a sufficiently long run, the supercooled droplet was crystallized into a polycrystalline fcc structure. The crystallization is a stochastic nucleation process. The time and the temperature of inception were evaluated over 42 samples, which indicate the existence of a stability limit.