Step 1: Recall biologically important coordination compounds.
Several vital biomolecules are coordination complexes: haemoglobin contains Fe$^{2+}$ at the centre of a porphyrin ring, and chlorophyll contains Mg$^{2+}$ at the centre of a chlorin ring.
Step 2: Use the name of Vitamin B$_{12}$ as a clue.
The systematic (chemical) name of Vitamin B$_{12}$ is cyanocobalamin. The root word "cobalamin" directly indicates that cobalt (Co) is the central metal atom in its structure.
Step 3: Determine the oxidation state.
In the corrin macrocycle of Vitamin B$_{12}$, cobalt is coordinated to four nitrogen atoms of the corrin ring and exists in the +3 oxidation state, giving the central ion Co$^{3+}$.
Step 4: Confirm the answer.
Vitamin B$_{12}$ is a coordination complex of cobalt(III), i.e., Co$^{3+}$, surrounded by a corrin ring.
\[ \boxed{Co^{3+}} \]