According to this paper the Bekenstein-Hawking entropy of a 1+1 black hole which described by the $SL_k(2,\mathbb{R})/U(1)$ WZW cigar geometry is given by the following formula appearing in eq. (5.7):
$$S=2\pi e^{-2\Phi_0}$$
But according to Bekenstein-Hawking formula the entropy is given by:
$$S_{BH}=\frac{k_Bc^3A}{4G\hbar}$$
We use natural units $c=\hbar=k_B=1$, So: $$S_{BH}=\frac{A}{4G}$$
The $d$ dimensional Newton constant $G$ is given by the string scale $l_s$ and string coupling $g_s$ in the following way: $$G_d=l_s^{d-2}g_s^2$$ for the 1+1 case we left with: $$G_2 = g_s^2.$$ Under dilaton field background, the coupling that measured from infinity is given by the asymptotic value of the dilaton $\Phi$ which we denote by $\Phi_0$ so: $$g_s=e^{\Phi_0}$$ Now, togather with a surface area of a 1d spatial line that is given by: $$A=2$$ The entropy should be:
$$S_{BH}=\frac{1}{2}e^{-2\Phi_0}$$
What is the source of the extra $4\pi$ factor such that $S_{BH}$ will fit exactly $S$?