  The zero-temperature phase diagram of the spin-$\frac{1}{2}$
$J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on an $AA$-stacked square-lattice
bilayer is studied using the coupled cluster method implemented to very high
orders. Both nearest-neighbor (NN) and frustrating next-nearest-neighbor
Heisenberg exchange interactions, of strengths $J_{1}>0$ and $J_{2} \equiv
\kappa J_{1}>0$, respectively, are included in each layer. The two layers are
coupled via a NN interlayer Heisenberg exchange interaction with a strength
$J_{1}^{\perp} \equiv \delta J_{1}$. The magnetic order parameter $M$ (viz.,
the sublattice magnetization) is calculated directly in the thermodynamic
(infinite-lattice) limit for the two cases when both layers have
antiferromagnetic ordering of either the N\'{e}el or the striped kind, and with
the layers coupled so that NN spins between them are either parallel (when
$\delta < 0$) or antiparallel (when $\delta > 0$) to one another. Calculations
are performed at $n$th order in a well-defined sequence of approximations,
which exactly preserve both the Goldstone linked cluster theorem and the
Hellmann-Feynman theorem, with $n \leq 10$. The sole approximation made is to
extrapolate such sequences of $n$th-order results for $M$ to the exact limit,
$n \to \infty$. By thus locating the points where $M$ vanishes, we calculate
the full phase boundaries of the two collinear AFM phases in the
$\kappa$--$\delta$ half-plane with $\kappa > 0$. In particular, we provide the
accurate estimate, ($\kappa \approx 0.547,\delta \approx -0.45$), for the
position of the quantum triple point (QTP) in the region $\delta < 0$. We also
show that there is no counterpart of such a QTP in the region $\delta > 0$,
where the two quasiclassical phase boundaries show instead an ``avoided
crossing'' behavior, such that the entire region that contains the nonclassical
paramagnetic phases is singly connected.
