We study the mechanism of confinement via the formation of Abrikosov-Nielsen-Olesen vortices on the Higgs branch of N = 2 supersymmetric SU(2) gauge theory with massive fundamental matter. Higgs branch represents a limiting case of a superconductor of type I with vanishing Higgs mass. We show that in this limit vortices still exist although they become logarithmically "thick". Because of this the confining potential is not linear any longer. It behaves as L/log L with distance L between the confining heavy charges (monopoles). This new confining regime can occur only in supersymmetric theories. We also address the problem of the quantum stability of vortices. To this end we develop a string representation for a vortex and use it to argue that vortices remain stable.