  Transport and magnetic studies of PbTaSe$_2$ under pressure suggest existence
of two superconducting phases with the low temperature phase boundary at $\sim
0.25$ GPa that is defined by a very sharp, first order, phase transition. The
first order phase transition line can be followed via pressure dependent
resistivity measurements, and is found to be near 0.12 GPa near room
temperature. Transmission electron microscopy and x-ray diffraction at elevated
temperatures confirm that this first order phase transition is structural and
occurs at ambient pressure near $\sim 425$ K. The new, high temperature / high
pressure phase has a similar crystal structure and slightly lower unit cell
volume relative to the ambient pressure, room temperature structure. Based on
first-principles calculations this structure is suggested to be obtained by
shifting the Pb atoms from the $1a$ to $1e$ Wyckoff position without changing
the positions of Ta and Se atoms. PbTaSe$_2$ has an exceptionally pressure
sensitive, structural phase transition with $\Delta T_s/\Delta P \approx -
1700$ K/GPa near 4 K, this first order transition causes an $\sim 1$ K ($\sim
25 \%$) step - like decrease in $T_c$ as pressure is increased through 0.25
GPa.
