We report the orbital distribution of the trans-Neptunian objects (TNOs) discovered during the Canada-France Ecliptic Plane Survey (CFEPS), whose discovery phase ran from early 2003 until early 2007. The follow-up observations started just after the first discoveries and extended until late 2009. We obtained characterized observations of 321 deg2 of sky to depths in the range g ~ 23.5-24.4 AB mag. We provide a database of 169 TNOs with high-precision dynamical classification and known discovery efficiency. Using this database, we find that the classical belt is a complex region with sub-structures that go beyond the usual splitting of inner (interior to 3:2 mean-motion resonance MMR), main (between 3:2 and 2:1 MMR), and outer (exterior to 2:1 MMR). The main classical belt (a = 40-47 AU) needs to be modeled with at least three components: the 'hot' component with a wide inclination distribution and two 'cold' components (stirred and kernel) with much narrower inclination distributions. The hot component must have a significantly shallower absolute magnitude (Hg ) distribution than the other two components. With 95% confidence, there are 8000+1800 --1600 objects in the main belt with Hg <= 8.0, of which 50% are from the hot component, 40% from the stirred component, and 10% from the kernel; the hot component's fraction drops rapidly with increasing Hg . Because of this, the apparent population fractions depend on the depth and ecliptic latitude of a trans-Neptunian survey. The stirred and kernel components are limited to only a portion of the main belt, while we find that the hot component is consistent with a smooth extension throughout the inner, main, and outer regions of the classical belt; in fact, the inner and outer belts are consistent with containing only hot-component objects. The Hg <= 8.0 TNO population estimates are 400 for the inner belt and 10,000 for the outer belt to within a factor of two (95% confidence). We show how the CFEPS Survey Simulator can be used to compare a cosmogonic model for the orbital element distribution to the real Kuiper Belt.