  Uncertainties in the thermonuclear rates of the
$^{15}$O($\alpha,\gamma$)$^{19}$Ne and $^{18}$F($p,\alpha$)$^{15}$O reactions
affect model predictions of light curves from type I X-ray bursts and the
amount of the observable radioisotope $^{18}$F produced in classical novae,
respectively. To address these uncertainties, we have studied the nuclear
structure of $^{19}$Ne over $E_{x} = 4.0 - 5.1$ MeV and $6.1 - 7.3$ MeV using
the $^{19}$F($^{3}$He,t)$^{19}$Ne reaction. We find the $J^{\pi}$ values of the
4.14 and 4.20 MeV levels to be consistent with $9/2^{-}$ and $7/2^{-}$
respectively, in contrast to previous assumptions. We confirm the recently
observed triplet of states around 6.4 MeV, and find evidence that the state at
6.29 MeV, just below the proton threshold, is either broad or a doublet. Our
data also suggest that predicted but yet unobserved levels may exist near the
6.86 MeV state. Higher resolution experiments are urgently needed to further
clarify the structure of $^{19}$Ne around the proton threshold before a
reliable $^{18}$F($p,\alpha$)$^{15}$O rate for nova models can be determined.
