The one-dimensional flame propagation in a gaseous mixture with a single one-step chemical reaction is considered. In a moving frame it is described by the equations \[ (1)\quad T_ t=T_{xx}+\Omega (T,Y)+V(t)T_ x,\quad Y_ t=(1/Le)Y_{xx}-\Omega (T,Y)+V(t)Y_ x, \] where: T is the temperature, Y is the mass fraction of the reactant, \(\Omega\) is the normalized reaction rate, Le is the Lewis number of the reactant, V(t) is the velocity of the moving frame with respect to the original fixed reference frame. In the proposed numerical scheme a finite-difference approximation on a moving mesh is used. The grid velocity V(t) is evaluated at each time step in such a way that the thermal energy contained in the computational domain is kept exactly constant. The discrete conservation of the variables is imposed for both the node displacements that occur at each time step and for the interpolations that are performed at some time levels, when a static grid adaption is done. In particular, an interpolation procedure is presented that is conservative and also has the advantages of preserving the positivity and monotonicity of the interpolated variables.