The diffusiophoretic motion of a charged spherical particle in an unbounded solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is analytically studied. The electrokinetic equations which govern the electric potential profile, the ionic concentration distributions (or electrochemical potential energies), and the velocity field in the fluid phase surrounding the particle are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved for a rigid dielectric sphere with its surface charge density (or zeta potential) as the small perturbation parameter. An analytical expression for the diffusiophoretic velocity of the colloidal particle in closed form is obtained from a balance between its electrostatic and hydrodynamic forces. This expression, which is correct to the second order of the surface charge density or zeta potential of the particle, is valid for an arbitrary value of kappa a, where kappa is the reciprocal of the Debye screening length and a is the particle radius. Our results agree well with the numerical solution in the literature for dielectric spheres with zeta potential up to 50 mV in 1:1 electrolytes.