Marginal convection in the form of Alfvén waves in an electrically conducting Bénard layer in the presence of a vertical magnetic field is investigated analytically using the Boussinesq model for the fluid. Small amplitude solutions are studied using the linearized magnetoconvection equations. These solutions are represented by double expansions in terms of two small parameters: a dimensionless viscosity and a dimensionless magnetic diffusivity. The leading-order problem corresponds to undamped Alfvén waves propagating between the boundaries of the fluid; buoyancy forces appear at higher order and can maintain the Alfvén waves against viscous and ohmic damping. The structure of the Alfvén waves is strongly dependent, even at leading order, on the physical nature of the walls. Four different types of boundary conditions are considered here: (A) illustrative, i.e. mathematically simple conditions, (B) solid, perfectly conducting walls, (C) vacuum external to the layer, and (D) solid, perfectly insulating walls. It is shown how in each case Alfvén waves are excited by a small, but sufficiently strong, thermal buoyancy but that, because of boundary layers, the solutions for the four sets of boundary conditions are very different.