Jupiter-family comets (JFCs) are the evolutionary products of trans-Neptunian objects (TNOs) that evolve through the giant planet region as Centaurs and into the inner solar system. Through numerical orbital evolution calculations following a large number of TNO test particles that enter the Centaur population, we have identified a short-lived dynamical Gateway, a temporary low-eccentricity region exterior to Jupiter through which the majority of JFCs pass. We apply an observationally based size distribution function to the known Centaur population and obtain an estimated Gateway region population. We then apply an empirical fading law to the rate of incoming JFCs implied by the the Gateway region residence times. Our derived estimates are consistent with observed population numbers for the JFC and Gateway populations. Currently, the most notable occupant of the Gateway region is 29P/Schwassmann-Wachmann 1 (SW1), a highly active, regularly outbursting Centaur. SW1's present-day, very-low-eccentricity orbit was established after a 1975 Jupiter conjunction and will persist until a 2038 Jupiter conjunction doubles its eccentricity and pushes its semi-major axis out to its current aphelion. Subsequent evolution will likely drive SW1's orbit out of the Gateway region, perhaps becoming one of the largest JFCs in recorded history. The JFC Gateway region coincides with a heliocentric distance range where the activity of observed cometary bodies increases significantly. SW1's activity may be typical of the early evolutionary processing experienced by most JFCs. Thus, the Gateway region, and its most notable occupant SW1, are critical to both the dynamical and physical transition between Centaurs and JFCs.