  We constrain a primordial magnetic field (PMF) generated during a phase
transition (PT) using the big bang nucleosynthesis bound on the relativistic
energy density. The amplitude of the PMF at large scales is determined by the
shape of the PMF spectrum outside its maximal correlation length scale. Even if
the amplitude of the PMF at 1 Mpc is small, PT-generated PMFs can leave
observable signatures in the potentially detectable relic gravitational wave
background if a large enough fraction ($1-10%$) of the thermal energy is
converted into the PMF.
