Benjamin recently put forward a model equation for the evolution of waves on the interface of a two-layer system of fluids in which surface tension effects are not negligible. In this case, the fluid motion $\eta$ on the interface of these two fluids can be approximately described by an equation
\eta_t+\eta_x+\eta\eta_x-\alpha L\eta_x\pm\beta\eta_{xxx}=0,
where $\eta$ depends on saptial variable $x$ and time variable $t$, and $L=H\partial_x$ is the composition of the Hilbert transform and the spatial derivative in the direction of primary propagation, or, equivalently, $L$ is a Fourier multiplier operator with symbol $|\xi|$. It is our purpose here to investigate the solitary-wave solutions of Benjamin's model. For a class of equations that include Benjamin's equation, which feature conflicting contributions to dispersion from dynamic effects on the interface and surface tension, we establish existence of travelling-wave solutions. This is complished by using P.L. Lions concentrated-compactness principle. Using the recently developed theory of Li and Bona, we are also able to determine rigorously the spatial asymptotics of these solutions.