We present a new design of the hat-fed reflector antenna for satellite communications, where a low reflection coefficient, high gain, low sidelobes and low cross-polar level are required over a wide frequency band. The hat feed has been optimized by using the Genetic Algorithm through a commercial FDTD solver, QuickWave-V2D, together with an own developed optimization code. The Gaussian vertex plate has been applied at the center of the reflector in order to improve the reflection coefficient and reduce the far-out sidelobes. A parabolic reflector with a ring-shaped focus has been designed for obtaining nearly 100% phase efficiency. The antenna's reflection coefficient is below -17 dB and the radiation patterns satisfy the M-x standard co- and cross-polar sidelobe envelopes for satellite ground stations over a bandwidth of 30%. A low-cost monolayer radome has been designed for the antenna with satisfactory performance. The simulations have been verified by measurements; both of them are presented in the paper.