  We prove the following asymptotic behavior for solutions to the generalized
Becker-D\"oring system for general initial data: under a detailed balance
assumption and in situations where density is conserved in time, there is a
critical density $\rho_s$ such that solutions with an initial density $\rho_0
\leq \rho_s$ converge strongly to the equilibrium with density $\rho_0$, and
solutions with initial density $\rho_0 > \rho_s$ converge (in a weak sense) to
the equilibrium with density $\rho_s$. This extends the previous knowledge that
this behavior happens under more restrictive conditions on the initial data.
The main tool is a new estimate on the tail of solutions with density below the
critical density.
