  Background: The breakout from the hot Carbon-Nitrogen-Oxigen (CNO) cycles can
trigger the rp-process in type I x-ray bursts. In this environment, a
competition between $^{15}\text{O}(\alpha,\gamma){^{19}\text{Ne}}$ and the
two-proton capture reaction $^{15}\text{O}(2p,\gamma){^{17}\text{Ne}}$ is
expected.
  Purpose: Determine the three-body radiative capture reaction rate for
${^{17}\text{Ne}}$ formation including sequential and direct, resonant and
non-resonant contributions on an equal footing.
  Method: Two different discretization methods have been applied to generate
$^{17}$Ne states in a full three-body model: the analytical transformed
harmonic oscillator method and the hyperspherical adiabatic expansion method.
The binary $p$--$^{15}$O interaction has been adjusted to reproduce the known
spectrum of the unbound $^{16}$F nucleus. The dominant $E1$ contributions to
the $^{15}\text{O}(2p,\gamma){^{17}\text{Ne}}$ reaction rate have been
calculated from the inverse photodissociation process.
  Results: Three-body calculations provide a reliable description of $^{17}$Ne
states. The agreement with the available experimental data on $^{17}$Ne is
discussed. It is shown that the $^{15}\text{O}(2p,\gamma){^{17}\text{Ne}}$
reaction rates computed within the two methods agree in a broad range of
temperatures. The present calculations are compared with a previous theoretical
estimation of the reaction rate.
  Conclusions: It is found that the full three-body model provides a reaction
rate several orders of magnitude larger than the only previous estimation. The
implications for the rp-process in type I x-ray bursts should be investigated.
