  The triple alpha reaction is a key to $^{12}$C production and is expected to
occur in weakly-coupled, thermal plasmas as encountered in normal stars. We
investigate how Coulomb screening affects the structure of a system of three
alpha particles in such a plasma environment by precise three-body calculations
within the Debye-H\"uckel approximation. A three-alpha model that has the
Coulomb interaction modified in the Yukawa form is employed. Precise three-body
wave functions are obtained by a superposition of correlated Gaussian bases
with the aid of the stochastic variational method. The energy shifts of the
Hoyle state due to the Coulomb screening are obtained as a function of the
Debye screening length. The results, which automatically incorporate the finite
size effect of the Hoyle state, are consistent with the conventional result
based on the Coulomb correction to the chemical potentials of ions that are
regarded as point charges in a weakly-coupled, thermal plasma. We have given a
theoretical basis to the conventional point-charge approach to the Coulomb
screening problem relevant for nuclear reactions in normal stars by providing
the first evaluation of the Coulomb corrections to the $Q$ value of the triple
alpha process that produces a finite size Hoyle state.
