  This work presents a first time accurate calculation of the magnetic dipole
hyperfine structure constants for the ground state and some low-lying excited
states of Pb$^+$. By comparing different levels of approximation with
experimental results, we demonstrate the importance of correlation effects
which reach beyond lower order relativistic many body perturbation theory.
Employing relativistic coupled-cluster theory we obtain a quantitative
understanding of the core-polarization and correlation effects inherent in this
system and observe completely different trends compared to $Ba^+$.
