We present a diffuse-interface model for the simulation of low Mach number multicomponent liquid-vapor flows with phase transition. Applications are for instance evaporating and boiling flows with weak compressibility effects. We start from a four-equation compressible two-phase flow model that assumes kinetic, mechanical and thermal equilibrium between the phases, and which includes viscous effects, heat conduction, surface tension, and mass transfer. By using the classical technique of asymptotic expansions in powers of a reference Mach number, we derive a two-phase low Mach number model suited for weakly compressible flows. A multicomponent version of the model is then introduced to describe flows composed of an incompressible liquid and a weakly compressible gaseous phase consisting of vapor and inert gas. Mass transfer in the model is driven by the difference of liquid and vapor chemical potentials. For the numerical solution of the low Mach number liquid-vapor-gas model we employ a finite volume scheme on a staggered grid based on a pressure correction method that uses a FFT-based Poisson solver. To prevent numerical diffusion of liquid-gas interfaces, regularization terms are included in the model system for interface sharpening. The proposed method ensures discrete conservation of the masses of the phases, and of the momentum and enthalpy of the flow system. We present numerical results of droplet evaporation tests, including a three-dimensional simulation of a sedimenting evaporating droplet.