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($beta palpha$)$^{15}$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($beta pgamma$)$^{19}$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.
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