  The double sine-Gordon field theory in the weak confinement regime is
studied. It represents the small non-integrable deformation of the standard
sine-Gordon model caused by the cosine perturbation with the frequency reduced
by the factor of 2. This perturbation leads to the confinement of the
sine-Gordon solitons, which become coupled into the 'meson' bound states. We
classify the meson states in the weak confinement regime, and obtain three
asymptotic expansions for their masses, which can be used in different regions
of the model parameters. It is shown, that the sine-Gordon breathers, slightly
deformed by the perturbation term, transform into the mesons upon increase of
the sine-Gordon coupling constant.
