Summary: We examine the linear stability of a premixed flame attached to a porous plug burner, using activation energy asymptotics. A dispersion relation is obtained which defines the stability boundaries in the wave-, Lewis- number plane, and the movement of these boundaries is followed as the mass flux is reduced below the adiabatic value and the flame moves towards the burner from infinity. Cellular instability is suppressed by the burner, but the pulsating instability usually associated with Lewis numbers greater than 1 is, at first, enhanced. For some parameter values the flame is never stable for all wavenumbers; the Lewis number stability band that exists for the unbounded flame disappears. For sufficiently small values of the stand-off distance the pulsating instability is suppressed.