  We obtain a one-parameter family $$(u_{\mu}(x,t),\eta_{\mu}(x,t))_{\mu\geq
\mu_0}=(\phi_{\mu}(x-\omega_{\mu} t),\psi_{\mu}(x-\omega_{\mu} t))_{\mu\geq
\mu_0}$$ of traveling-wave solutions to the Boussinesq system
  $$u_t+\eta_x+uu_x+c\eta_{xxx}=0,\eta_t+u_x+(\eta u)_x+au_{xxx}=0$$ in the
case $a,c<0$, with non-null speeds $\omega_{\mu}$ arbitrarily close to $0$
($\omega_{\mu}\xrightarrow[\mu\to+\infty]{} 0$). We show that the $L^2$-size of
such traveling-waves satisfies the uniform (in $\mu$) estimate
$\|\phi_{\mu}\|_2^2+\|\psi_{\mu}\|_2^2\leq C\sqrt{|a|+|c|},$ where $C$ is a
positive constant. Furthermore, $\phi_{\mu}$ and $-\psi_{\mu}$ are smooth,
non-negative, radially decreasing functions which decay exponentially at
infinity.
