  The conversion between the formal and observed parameters is obtained from
the phase equivalence between the $R$-matrix and Breit-Wigner formula, and it
is applied to a study of the low-energy $E$1 $S$-factor of
$^{12}$C($\alpha$,$\gamma_0$)$^{16}$O. As an example, weak interference between
1$^-$ states in $^{16}$O is discussed. As well as the ordinary $R$-matrix
method, the present method calculates the $S$-matrix and cross sections,
independent of the boundary condition. Therefore, the $E$1 $S$-factor at
$E_{c.m.} = 300$ keV is found to be reduced from the current evaluation.
