  Background: The $^{15}$O($\alpha ,\gamma$)$^{19}$Ne bottleneck reaction in
Type I x-ray bursts is the most important thermonuclear reaction rate to
constrain experimentally, in order to improve the accuracy of burst light-curve
simulations. A proposed technique to determine the thermonuclear rate of this
reaction employs the $^{20}$Mg($\beta p\alpha$)$^{15}$O decay sequence. The key
$^{15}$O($\alpha ,\gamma$)$^{19}$Ne resonance at an excitation of 4.03 MeV is
now known to be fed in $^{20}$Mg($\beta p\gamma$)$^{19}$Ne; however, the
energies of the protons feeding the 4.03 MeV state are unknown. Knowledge of
the proton energies will facilitate future $^{20}$Mg($\beta p \alpha$)$^{15}$O
measurements.
  Purpose: To determine the energy of the proton transition feeding the 4.03
MeV state in $^{19}$Ne.
  Method: A fast beam of $^{20}$Mg was implanted into a plastic scintillator,
which was used to detect $\beta$ particles. 16 high purity germanium detectors
were used to detect $\gamma$ rays emitted following $\beta p$ decay. A Monte
Carlo method was used to simulate the Doppler broadening of $^{19}$Ne $\gamma$
rays and compare to the experimental data.
  Results: The center of mass energy between the proton and $^{19}$Ne, feeding
the 4.03 MeV state, is measured to be 1.21${^{+0.25}_{-0.22}}$ MeV,
corresponding to a $^{20}$Na excitation energy of 7.44${^{+0.25}_{-0.22}}$ MeV.
Absolute feeding intensities and $\gamma$-decay branching ratios of $^{19}$Ne
states were determined including the 1615 keV state. A new $\gamma$ decay
branch from the 1536 keV state in $^{19}$Ne to the ground state is reported.
The lifetime of the 1507 keV state in $^{19}$Ne is measured to be
4.3${^{+1.3}_{-1.1}}$ ps resolving discrepancies in the literature. Conflicting
$^{20}$Mg($\beta p$) decay schemes in published literature are clarified.
