Several diffusion-dominated problems in science and engineering require the solution of Poisson-type  equations in irregular domains. These types of problems include free-boundary (Stefan-type) problems that describe interesting physical phenomena such as the crystallization and solidification of different materials, where the evolution of the interface is not known a priori, and usually depends on the gradients of the solution. Therefore, it is important to develop numerical methods for solving Poisson-type equations that produce not only accurate solutions, but also accurate gradients of the solutions. In this work, we present a numerical approach for solving the Poisson equation in irregular domains with Robin boundary conditions. We employ the level-set method to represent irregular domains, and discretize the Poisson equation using a combination of a classical, second-order discretization for the internal nodes and first-order finite-volume discretization for the interfacial nodes. We demonstrate in several examples that our method is second-order-accurate in both the solution and the gradients of the solution.