Magnetic thin films with perpendicular magnetic anisotropy (PMA) have localized excitations that correspond to reversed dynamically precessing magnetic moments, known as magnetic droplet solitons. Fundamentally, these excitations are associated with an attractive interaction between elementary spin-excitations (i.e., magnons) and were predicted to occur in PMA materials in the absence of damping [1,2]. While damping, present in all magnetic materials, suppresses these excitations, it is now possible to compensate damping by spin transfer torques through electrical current flow in nanometer scale contacts to ferromagnetic thin films [3,4]. A theory predicts the appearance of magnetic droplet solitons at a threshold current in nanocontacts  and, recently, experimental signatures of droplet nucleation have been reported . However, thus far, they have been observed to be nearly reversible excitations, with only partially reversed magnetization and to be subject to instabilities that cause them to drift away from the nanocontacts (i.e., drift instabilities) . Here we show that magnetic droplet solitons can be stabilized in a spin transfer nanocontact. Further, they exhibit a strong hysteretic response to fields and currents and a nearly fully reversed magnetization in the contact. These observations, in addition to their fundamental interest, open up new applications for magnetic droplet solitons as multi-state high frequency current and field tunable oscillators.