We measure the transmission spectrum of an optical cavity filled with
two-level atoms. For small intensities, the coupling between atoms and
a single mode of the cavity splits the spectrum into two normal modes
: the vacuum Rabi peaks. We gradually decouple the system in two ways
and study the evolution of the vacuum Rabi peaks. First we lower the a
tom-cavity coupling frequency through its intensity dependence, showin
g that the peaks shift, deform, exhibit frequency hysteresis, and even
tually merge into a sin:le peak that approaches the empty cavity reson
ance. Second we detune the atoms and the cavity mode and observe an av
oided crossing. The normal modes of the coupled system transform into
the two resonances of the uncoupled system. We map out the transition
region and find good agreement with theoretical predictions.