  We study the supersymmetrized version of the singlet majoron model and,
performing an analysis of the renormalization group equation improved
potential, we find that a spontaneous breaking of $R$-parity can be achieved
for a wide range of the parameters. Studying the finite temperature effective
potential, we show that the phase transition leading to $R$-parity breaking can
be of the first order and can occur at temperatures below the weak scale, thus
avoiding any constraint coming from the requirement of the preservation of the
baryon asymmetry in the early Universe.
