  We use the coupled cluster method to study the zero-temperature quantum phase
diagram of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on
the honeycomb bilayer lattice. In each layer we include both nearest-neighbor
and frustrating next-nearest-neighbor antiferromagnetic exchange couplings, of
strength $J_{1}>0$ and $J_{2} \equiv \kappa J_{1} > 0$, respectively. The two
layers are coupled by an interlayer nearest-neighbor exchange, with coupling
constant $J_{1}^{\perp} \equiv \delta J_{1}>0$. We calculate directly in the
infinite-lattice limit both the ground-state energy per spin and the N\'{e}el
magnetic order parameter, as well as the triplet spin gap. By implementing the
method to very high orders of approximation we obtain an accurate estimate for
the full boundary of the N\'{e}el phase in the $\kappa\delta$ plane. For each
value $\delta < \delta_{c}^{>}(0) \approx 1.70(5)$ we find an upper critical
value $\kappa_{c}(\delta)$, such that N\'{e}el order is present for $\kappa <
\kappa_{c}(\delta)$. Conversely, for each value $\kappa < \kappa_{c}(0) \approx
0.19(1)$ we find an upper critical value $\delta_{c}^{>}(\kappa)$, such that
N\'{e}el order persists for $0 < \delta < \delta_{c}^{>}(\kappa)$. Most
interestingly, for values of $\kappa$ in the range $\kappa_{c}(0) < \kappa <
\kappa^{>} \approx 0.215(2)$ we find a reentrant behavior such that N\'{e}el
order exists only in the range $\delta_{c}^{<}(\kappa) < \delta <
\delta_{c}^{>}(\kappa)$, with $\delta_{c}^{<}(\kappa)>0$. These latter upper
and lower critical values coalesce when $\kappa = \kappa^{>}$, such that
$\delta_{c}^{<}(\kappa^{>}) = \delta_{c}^{>}(\kappa^{>}) \approx 0.25(5)$.
