We have constructed merger trees for galaxies in the Illustris Simulation by directly tracking the baryonic content of subhalos. These merger trees are used to calculate the galaxy-galaxy merger rate as a function of descendant stellar mass, progenitor stellar mass ratio, and redshift. We demonstrate that the most appropriate definition for the mass ratio of a galaxy-galaxy merger consists in taking both progenitor masses at the time when the secondary progenitor reaches its maximum stellar mass. Additionally, we avoid effects from `orphaned galaxies by allowing some objects to `skip a snapshot when finding a descendant, and by only considering mergers which show a well-defined `infall moment. Adopting these definitions, we obtain well-converged predictions for the galaxy-galaxy merger rate with the following main features, which are qualitatively similar to the halo-halo merger rate except for the last one: a strong correlation with redshift that evolves as $sim (1+z)^{2.4-2.8}$, a power law with respect to mass ratio, and an increasing dependence on descendant stellar mass, which steepens significantly for descendant stellar masses greater than $sim 2 times 10^{11} , {rm M_{odot}}$. These trends are consistent with observational constraints for medium-sized galaxies ($M_{ast} gtrsim 10^{10} , {rm M_{odot}}$), but in tension with some recent observations of the close pair fraction for massive galaxies ($M_{ast} gtrsim 10^{11} , {rm M_{odot}}$), which report a nearly constant or decreasing evolution with redshift. Finally, we provide a fitting function for the galaxy-galaxy merger rate which is accurate over a wide range of stellar masses, progenitor mass ratios, and redshifts.
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