A strongly driven cavity containing a single resonant strongly coupled atom exhibits a phase bistability. The phase of the field is strongly correlated with the phase of the atomic dipole. It has been shown previously that phase-sensitive monitoring of the field emitted by the cavity would induce conditional quantum jumps between orthogonal atomic dipole states ("dressed" states). Here we show that such monitoring can be used to fix the atom into a single dressed state. As soon as a state-changing quantum jump is inferred from the measurement of the field, the atomic state is flipped using a p pulse. We study this feedback scheme analytically and numerically. We show that the occupation probability of the desired fixed state can be as high as 1-1/8?C1, where C1뱠is the single-atom cooperativity and ? the detection efficiency (which does not have to be close to unity). The control of the atomic dynamics is manifest in the fluorescence spectrum. The widths of all three peaks are modified from the usual Mollow spectrum, and almost all of the area under one of the sidebands is transferred to the other sideband. This is as expected, as one of the dressed states is essentially unoccupied, and transitions out of it do not occur. In addition, the width of the central peak goes to zero. This indicates coherent scattering due to the nonzero mean atomic dipole created by the feedback.