The flow of giant lipid vesicles through cylindrical capillaries is experimentally investigated. Vesicles are deflated with reduced volumes between 0.8 and 1, corresponding to prolate spheroidal equilibrium shapes. Both interior and exterior fluids are sugar solutions with viscosities close to 10-3 Pa s. Vesicles are aspirated into a capillary tube with a diameter close to the vesicle size and a constant flow rate is imposed. Significant deformation of the membrane occurs and increases when the velocity, confinement or deflation of the vesicle are increased. The mobility of vesicles, defined as the ratio of their velocity to the average velocity of the fluid is a decreasing function of confinement. Our experimental system provides a controllable and flexible tool to investigate deformability effects responsible for crucial aspects of blood rheology in capillaries.