We construct oscillatory solitary wave solutions of a fifth-order Korteweg-de Vries equation, where the oscillations decay at infinity. These waves arise as a bifurcation from the linear dispersion curve at that wavenumber where the linear phase speed and group velocity coincide. Our approach is a wave-packet analysis about this wavenumber which leads in the first instance to a higher-order nonlinear Schrödinger equation, from which we then obtain the steady solitary wave solution. We then describe a complementary normal-form analysis which leads to the same result. In addition we derive the nonlinear Schrödinger equation for all wavenumbers, and list all the various anomalous cases.