CTB 109 (G109.1-1.0) is a Galactic supernova remnant (SNR) with a hemispherical shell morphology in X-rays and in the radio band. In this work, we report the detection of γ-ray emission coincident with CTB 109, using 37 months of data from the Large Area Telescope on board the Fermi Gamma-ray Space Telescope. We study the broadband characteristics of the remnant using a model that includes hydrodynamics, efficient cosmic-ray (CR) acceleration, nonthermal emission, and a self-consistent calculation of the X-ray thermal emission. We find that the observations can be successfully fit with two distinct parameter sets, one where the γ-ray emission is produced primarily by leptons accelerated at the SNR forward shock and the other where γ-rays pr...