The interplay between superconductivity and the charge-density wave (CDW) state in pure 1T-TiSe2 is examined through a high-pressure study extending up to pressures of 10 GPa between sub-Kelvin and room temperatures. At a critical pressure of 2 GPa a superconducting phase sets in and persists up to pressures of 4 GPa. The maximum superconducting transition temperature is 1.8 K. These findings complement the recent discovery of superconductivity in copper-intercalated 1T-TiSe2. The comparisons of the normal state and superconducting properties of the two systems reveal the possibility that the emergent electronic state qualitatively depends on the manner in which the CDW state is destabilized, making this a unique example where two different superconducting domes are obtained by two different methods from the same parent compound.