The behavior of an axisymmetric vertical turbulent jet in an unconfined stratified environment is studied by means of well-resolved, large-eddy simulations. The stratification is two uniform layers separated by a thermocline. This study considers two cases: when the thermocline thickness is small and on the order of the jet diameter at the thermocline entrance. The Froude number of the jet at the thermocline varies from 0.6 to 1.9, corresponding to the class of weak fountains. The mean jet penetration, stratified turbulent entrainment, jet oscillations, and the generation of internal waves are examined. The mean jet penetration is predicted well by a simple model based on the conservation of the source energy in the thermocline. The entrainment coefficient for the thin thermocline is consistent with the theoretical model for a two-layer stratification with a sharp interface, while for the thick thermocline entrainment is larger at low Froude numbers. The data reveal the presence of a secondary horizontal flow in the upper part of the thick thermocline, resulting in the entrainment of fluid from the thermocline rather than from the upper stratification layer. The spectra of the jet oscillations in the thermocline display two peaks, at the same frequencies for both stratifications at fixed Froude number. For the thick thermocline, internal waves are generated only at the lower frequency, since the higher peak exceeds the maximal buoyancy frequency. For the thin thermocline, conversely, the spectra of the internal waves show the two peaks at low Froude numbers, whereas only one peak at the lower frequency is observed at higher Froude numbers.