The problem of computing a d-dimensional Euclidean Voronoi diagram of spheres is relevant to many areas, including computer simulation, motion planning, CAD, and computer graphics. This paper presents a new algorithm based on the explicit computation of the coordinates and radii of Euclidean Voronoi diagram vertices for a set of spheres. The algorithm is further applied to compute the Voronoi diagram with a specified precision in a fixed length floating-point arithmetic. The algorithm is implemented using the ECLibrary (Exact Computation Library) and tested on the example of a 3-dimensional Voronoi diagram of a set of spheres.