Brownian coagulation is an important particle growth mechanism in a number of aerosol processes. Analytical solution to Brownian coagulation has been an interesting subject in aerosol population dynamics field because it can provide deep insight into the nature of the coagulation process. In the case of spherical particles, analytical solutions have been derived for all different particle size regimes. For fractal aggregates that can represent the morphology of many important industrial aerosol products and combustion soot particles, however, analytical solutions have been reported only for the free-molecule and continuum regimes. In this work, a semi-analytical solution to the Brownian coagulation of fractal aggregates valid for the entire particle size range is presented using the method of moments based on the assumption of log-normal particle size distribution. The new solution is validated by limiting-case analyses as well as by comparison with a more accurate numerical solution. The characteristics of the Brownian coagulation of fractal aggregates in the transition regime are discussed using the new solution.