During exocytosis secretory vesicles fuse with a target membrane and release neurotransmitters, hormones or other bioactive molecules through a membrane fusion pore. The initially small pore may subsequently dilate for full contents release, as commonly observed in amperometric traces. The size, shape and evolution of the pore is critical to the course of contents release, but exact fusion pore solutions accounting for membrane tension and bending energy constraints have not been available. Here we obtained exact solutions for fusion pores between two membranes. We find three families: a narrow pore, a wide pore and an intermediate tether-like pore. For high tensions these are close to the catenoidal and tether solutions recently reported for freely hinged membrane boundaries. We suggest membrane fusion initially generates a stable narrow pore, and for high enough membrane tension pore dilation represents a transition from the narrow pore to the stable wide pore family. The unstable intermediate pore is a transition state setting the energy barrier for pore dilation. Finally, we study fusion pores in nanodiscs, powerful systems for the study of individual pores. We show that nanodiscs stabilize fusion pores by locking them into the small pore family.