The phenomenon of enhanced backscattering in the scattering of light from a randomly rough surface is the presence of a well-defined peak in the retroreflection direction in the angular dependence of the intensity of the light scattered diffusely from the surface. A striking feature of this phenomenon is that it occurs for any angle of incidence. Suppose, however, that one would like to have a random surface that displays enhanced backscattering for only a single, specified, angle of incidence. Such a surface could be useful, for example, in situations where one wishes to position a source (and hence the detector) at a specified direction with respect to the site at which the scattering surface is situated. In this paper we show how a one-dimensional random surface can be generated that produces an enhanced backscattering peak for only a specified angle of incidence when illuminated by p-polarized light whose plane of incidence is perpendicular to the generators of the surface. This surface is defined by a power spectrum (the Fourier transform of the surface height autocorrelation function) given by g(Q) = (π)/(4(Δ)k)[θ (Q-k1+Δk)θ(k1+Δk-Q)+θ(Q-k2+Δk-Q)θ (k2+Δk-Q)+θ(-Q-k1+Δk)θ (k1+Δk+Q)+θ(-Q-k2+Δk)θ (k2+Δk+Q)], where θ(z) is the Heaviside unit step function, k1= kR-k0,k2=kR-k0, k(subscript R is the real part of the wavenumber of the surface plasmon polariton of frequency ω supported by the planar vacuum-metal interface, and k0 is related to the angle of incidence measured clockwise from the x3-axis by k0=(ω/c)sinθ0. An explanation is provided for why a surface defined by this power spectrum produces enhanced backscattering at only the angle of incidence given by θs=-θ0, and it is confirmed by numerical calculations of the angular dependence of the intensity of the light scattered diffusely from it.