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Magnetic helicity is a quantity of great importance in solar studies because it is conserved in ideal magnetohydrodynamics. While many methods for computing magnetic helicity in Cartesian finite volumes exist, in spherical coordinates, the natural coordinate system for solar applications, helicity is only treated approximately. We present here a method for properly computing the relative magnetic helicity in spherical geometry. The volumes considered are finite, of shell or wedge shape, and the three-dimensional magnetic field is considered to be fully known throughout the studied domain. Testing of the method with well-known, semi-analytic, force-free magnetic-field models reveals that it has excellent accuracy. Further application to a set of nonlinear force-free reconstructions of the magnetic field of solar active regions and comparison with an approximate method used in the past indicates that the proposed method can be significantly more accurate, thus making our method a promising tool in helicity studies that employ spherical geometry. Additionally, we determine and discuss the applicability range of the approximate method. [ABSTRACT FROM AUTHOR]