The structural basis for the aggregation-induced optical properties of colloidal gold nanosphere aggregates is examined by means of electrodynamics calculations. Recently developed methods for calculating the electrodynamic response of aggregates composed of large numbers of small metal nanospheres in a dielectric medium are used to determine the optical changes associated with the formation of spherical aggregates. The calculations use accurate nanoparticle polarizabilities determined from Mie theory, an iterative conjugate-gradient solution algorithm, and fast-Fourier transform methods for efficient solution of the electrodynamic interacting nanoparticle equations. The UV extinction lowering and the shifting and broadening of the visible plasmon peak observed experimentally in solutions of DNA-linked gold nanospheres are explained as the collective electromagnetic response of thousands of nanoparticles.