A one-dimensional model of a boiling superheated droplet is developed and analytically evaluated. The droplet is suspended initially at rest in an immiscible liquid, and boiling is initiated from a bubble nucleus located at its center. Numerical simulations of the boiling process are carried out for conditions corresponding to those typically found in boiling emulsions. The boiling process is found to proceed rapidly, but there is no indication that a shockwave forms, as has been speculated in some models of boiling in dilute emulsions. The rate of boiling of the droplet is influenced by the properties of the surrounding liquid most strongly near the end of the boiling process. The resulting bubble grows momentarily larger than its final equilibrium diameter and oscillates in diameter before reaching equilibrium.