We identify an acoustic process in which the conversion of angular momentum between its spin and orbital form takes place. The interaction between an evanescent wave propagating at the interface of two immiscible fluids and an isolated droplet is considered. The elliptical motion of the fluid supporting the incident wave is associated with a simple state of spin angular momentum, a quantity recently introduced for acoustic waves in the literature. We experimentally observe that this field predominantly forces a directional wave transport circling the droplet’s interior, revealing the existence of confined phase singularities. The circulation of the phase, around a singular point, is characteristic of angular momentum in its orbital form, thereby demonstrating the conversion mechanism. The numerical and experimental observations presented in this Letter have implications for the fundamental understanding of the angular momentum of acoustic waves, and for applications such as particle manipulation with radiation forces or torques, acoustic sensing and imaging.