We evaluate the field-dependent energy barriers associated with annihilation of a Neel skyrmion in ultrathin magnetic nanodots with interfacial Dzyaloshinskii-Moriya interaction. Three scenarios of the skyrmion annihilation are considered: expansion or contraction of the radius of the skyrmion and skyrmion escape through the dot boundary. For typical parameters of Co/Pt circular nanodots, we find that at zero magnetic field, the energy barrier associated with a displacement of the skyrmion core is always lower than that corresponding to its expansion or contraction. Importantly, the energy barrier value is small, so that the skyrmion in ultrathin dots is unstable versus thermal fluctuations at room temperature for a long time scale. However, we find that the skyrmion can be stabilized by an out-of-plane magnetic field applied parallel to its core. The increase of the energy barriers corresponds to a magnetic field interval, where skyrmions of small and large radius coexist and the large-core-radius skyrmion is more stable than the small one. In this field region, the skyrmion can be stable for hours before it is annihilated by thermal fluctuations escaping through the dot edge.