  We consider the nonlinear integrodifferential Benjamin-Bona-Mahony equation
$$ u_t - u_{txx} + u_x - \int_0^\infty g(s) u_{xx}(t-s) {\rm d} s + u u_x = f
$$ where the dissipation is entirely contributed by the memory term. Under a
suitable smallness assumption on the external force $f$, we show that the
related solution semigroup possesses the global attractor in the natural weak
energy space. The result is obtained by means of a nonstandard approach based
on the construction of a suitable family of attractors on certain invariant
sets of the phase space.
