  Using optical Ramsey interferometry, we precisely measure the laser-induced
AC-stark shift on the $S_{1/2}$ -- $D_{5/2}$ "quantum bit" transition near 729
nm in a single trapped $^{40}$Ca$^+$ ion. We cancel this shift using an
additional laser field. This technique is of particular importance for the
implementation of quantum information processing with cold trapped ions. As a
simple application we measure the atomic phase evolution during a $n \times
2\pi$ rotation of the quantum bit.
