Results for the linear thermoconvective stability of a layer of viscoelastic Maxwell fluid are presented. The stability problem is characterized by taking into account the lower and upper wall thermal conductivities as well as their thicknesses. This allows more realistic theoretical boundary conditions. A system consisting of a horizontal infinite Maxwell fluid layer confined between two parallel walls perpendicular to gravity is considered. The critical Rayleigh number R-c, the frequency of oscillation omega(c) and the wavenumber k(c) were determined for fixed values of the relaxation time constant F and the Prandtl number Pr. The results are given for a range of wall thermal conductivities and thicknesses. Analytical and numerical solutions were calculated. Some unexpected results were found in comparison to those of the Newtonian fluid where the criticality curves become more unstable when the conductivities of the walls change from very good conductors to very bad conductors. (C) 2011 Elsevier Ltd. All rights reserved.