  We develop one-loop effective vertices and propagators in the linear sigma
model at finite temperature satisfying the chiral Ward identities. We use these
in turn to compute the pion dispersion relation in a pion medium for small
momentum and temperatures on the order of the pion mass at next to leading
order in the parameter m_pi^2/4pi^2f_pi^2 and to zeroth order in the parameter
(m_pi/m_sigma)^2. We show that this expansion reproduces the result obtained
from chiral perturbation theory at leading order. The main effect is a
perturbative, temperature-dependent increase of the pion mass. We find no
qualitative changes in the pion dispersion curve with respect to the leading
order behavior in this kinematical regime.
