  Let $\mathcal{R}$ be a free Lie conformal algebra of rank $2$ with
$\mathbb{C}[\partial]$-basis $\{L,I\}$ and relations \begin{eqnarray*}
\left[L_{\lambda} L\right]=(\partial+2 \lambda) (L+I),\ \left[L_{\lambda}
I\right]=(\partial+\lambda) I, \ \left[I_{\lambda} L\right]=\lambda I,\
\left[I_{\lambda} I\right]=0. \end{eqnarray*} In this paper, we first classify
all finite nontrivial irreducible conformal modules over $\mathcal{R}$. Then we
determine extensions between two finite irreducible conformal
$\mathcal{R}$-modules.
