Let $M$ be a $C^\infty$-manifold with a Riemannian conformal structure $C$. Given a regular curve $\gamma$ on $M$, the authors define a linear operator on the space of (differentiable) vector fields along $\gamma$, only depending on $C$, called the Fermi-Walker connection along $\gamma$. Then, the authors introduce the concept of Fermi-Walker parallel vector field along $\gamma$, proving that such vector fields set up a linear space isomorphic to the tangent space at a point of $\gamma$. This allows to consider the Fermi-Walker horizontal lift of $\gamma$ to the bundle $CO(M)$ of conformal frames on $M$ and to define, for any conformal frame $b$ at a point $p$, a lift function $k_b$ from the set of 2-jets of regular curves on $M$ starting at $p$ into the tangent space $T_b(CO(M))$. Finally, using the lift functions $k_b$, $b\in CO(M) $, the authors construct a trivialization of the fiber bundle $CO(M)_1$ over $CO(M)$, $CO(M)_1$, denoting the first prolongation of !