Magnetic vortices can be stabilized in magnetic materials by a so-called Dzyaloshinsky interaction. Their structure is calculated systematically for uniaxial ferromagnetic materials of the easy-axis type by numerically solving the differential equations in the circular cell approximation. In reduced units two external parameters are left over: the value of an external field parallel to the crystal axis and the relative strength of the Dzyaloshinsky interaction. A phase diagram in these variables consists of three thermodynamically stable phases: a uniform state at high field values, a one-dimensionally modulated spiral state at low fields and the new vortex state in an intermediate field range. The corresponding calculated magnetization curves clearly show the transitions between these states.