The exact form of the solution for the vector potential and magnetic field of a rotating uniformly charged ball is explicitly found. Expressions for specific vector spherical polynomials associated with the corresponding components of the potential are used to represent the solution. Inside and outside a charged ball uniformly rotating around its axis, the components of the potential and magnetic field are determined up to terms containing the sixth power of the speed of light in the denominator. To do this, it was necessary to use spherical coordinates and eight polynomials of each type. In addition, the solutions inside and outside the ball were equated to each other on the surface of the ball, taking into account the symmetry of the ball. The accuracy of the approach used can be increased, since it is determined only by the number of polynomials used, whose contribution to solutions decreases rapidly as the degree of the polynomials increases. To calculate the vector potential and magnetic field of a rotating ball within the framework of special relativity, it is sufficient to substitute the coordinates of the observation point, the invariant volumetric charge density, the angular velocity of rotation and the radius of the ball into the formulas.