We observed comet 2P/Encke with the Infrared Space Observatory ISOCAM on July 14, 1997 from a particularly favorable viewing geometry above the comet's orbital plane and at a distance of 0.25 AU. A structured coma was observed, along with a long, straight dust trail. For the first time, we are able to observe the path of particles as they evolve from the nucleus to the trail. The particles that produce the infrared coma are large, with a radiation to gravitational force ratio beta<0.001 (corresponding to >mm-sized particles). The dust trail follows the orbit of the comet across our image, with a central core that is 20,000 km wide, composed of particles with beta<1e-5 (size \(\sim 5\) cm) from previous apparitions. The abundant large particles near the comet pose a significant hazard to spacecraft. There is no evidence of a classical cometary dust tail due to small particles with beta>0.001, in marked contrast to other comets like P/Halley or C/Hale-Bopp. The structure of the coma requires anisotropic emission and that the spin axis of the nucleus to be nearly parallel to the orbital plane, resulting in strong seasonal variations of the particle emission. While most of the infrared coma emission is due to dust produced during the 1997 apparition, the core of the dust trail requires emissions from previous apparitions. The total mass lost during the 1997 apparition is estimated to be 2-6e13 g. Comparing to the gas mass loss from ultraviolet observations, the dust-to-gas mass ratio is 10-30, much higher than has ever been suggested from visual light observations. Using the recently-measured nuclear diameter, we find that Encke can only last 3000-10,000 rhoN yr (where rhoN is the nuclear density in g/cc) at its present mass loss rate.