  We study the cosmological history of the classical conformal $B-L$ gauge
extension of the standard model, in which the physical scales are generated via
the Coleman-Weinberg-type symmetry breaking. Especially, we consider the
thermal phase transition of the U$(1)_{B-L}$ symmetry in the early universe and
resulting gravitational-wave production. Due to the classical conformal
invariance, the phase transition tends to be a first-order one with
ultra-supercooling, which enhances the strength of the produced gravitational
waves. We show that, requiring (1) U$(1)_{B-L}$ is broken after the reheating,
(2) the $B-L$ gauge coupling does not blow up below the Planck scale, (3) the
thermal phase transition completes in almost all the patches in the universe,
the gravitational wave spectrum can be as large as $\Omega_{\rm GW} \sim
10^{-8}$ at the frequency $f \sim 0.01$-$1$Hz for some model parameters, and a
vast parameter region can be tested by future interferometer experiments such
as eLISA, LISA, BBO and DECIGO.
