A nonlinear model of diffusion-limited growth of three-dimensional colloidal aggregate has been developed. The conditions of internal combination/recombination balance has been considered. Obtained solution shows that the aggregate contains the central dense core and the surrounding loose region, in which the aggregated particle concentration decreases as a power function, according to the fractal cluster properties. The fractal dimension d(f) has been found to be independent on physical and chemical properties of colloidal system. The obtained universal value d(f)=2.5 is in a good agreement with the known experimental and numerical results. The model may be useful for the analysis of a wide class of aggregation phenomena.