  We study the nonresonant part of the $^7$Be($p,\gamma$)$^8$B reaction using a
three-cluster resonating group model that is variationally converged and
virtually complete in the $^4$He+$^3$He+$p$ model space. The importance of
using adequate nucleon-nucleon interaction is demonstrated. We find that the
low-energy astrophysical $S$-factor is linearly correlated with the quadrupole
moment of $^7$Be. A range of parameters is found where the most important
$^7$Be and $^7$Li properties are reproduced simultaneously; the corresponding
$S$-factor at $E_{\rm cm}=20$ keV is $24.6-26.1$ eV$\cdot$b.
