We develop a new approach to Gelfand-Zeitlin theory for the symplectic group $Sp(n,\mathbb{C})$. Classical Gelfand-Zeitlin theory, concerning $GL(n,\mathbb{C})$, rests on the fact that branching from $GL(n,\mathbb{C})$ to $GL(n-1,\mathbb{C})$ is multiplicity-free. Since branching from $Sp(n,\mathbb{C})$ to $Sp(n-1,\mathbb{C})$ is not multiplicity-free, the theory cannot be directly applied to this case. Let $L$ be the $n$-fold product of $SL(2,\mathbb{C})$. Our main theorem asserts that each multiplicity space that arises in the restriction of an irreducible representation of $Sp(n,\mathbb{C})$ to $Sp(n-1,\mathbb{C}$, has a unique irreducible $L$-action satisfying certain naturality conditions. We also given an explicit description of the $L$-module structure of each multiplicity space. As an application we obtain a Gelfand-Zeitlin type basis for the irreducible representations of $Sp(n,\mathbb{C})$.