  A spectrum exhibiting $E_8$ symmetry is expected to arise when a small
longitudinal field is introduced in the transverse-field Ising chain at its
quantum critical point. Evidence for this spectrum has recently come from
neutron scattering measurements in cobalt niobate, a quasi one-dimensional
Ising ferromagnet. Unlike its zero-temperature counterpart, the
finite-temperature dynamics of the model has not yet been determined. We study
the dynamical spin structure factor of the model at low frequencies and nonzero
temperatures, using the form factor method. Its frequency dependence is
singular, but differs from the diffusion form. The temperature dependence of
the nuclear magnetic resonance (NMR) relaxation rate has an activated form,
whose prefactor we also determine. We propose NMR experiments as a means to
further test the applicability of the $E_8$ description for CoNb$_2$O$_6$.
