For $m\in \mathbb{N}$, we determine the irreducible components of the $m-$th Jet Scheme of a complex branch $C$ and we give formulas for their number $N\left(m\right)$ and for their codimensions, in terms of $m$ and the generators of the semigroup of $C$. This structure of the Jet Schemes determines and is determined by the topological type of $C$.