We present a joint experimental-theoretical study on the effect of the carrier-envelope phase (CEP) of a few-cycle pulse on the atomic excitation process. We focus on the excitation rates of argon as a function of CEP in the intensity range from 50-300 TW/cm$^2$, which covers the transition between the multiphoton and tunneling regimes. Through numerical simulations based on solving the time-dependent Schr{o}dinger equation (TDSE), we show that the resulting bound-state population is highly sensitive to both the intensity and the CEP. Because the intensity varies over the interaction region, the CEP effect is considerably reduced in the experiment. Nevertheless, the data clearly agree with the theoretical prediction, and the results encourage the use of precisely tailored laser fields to coherently control the strong-field excitation process. We find a markedly different behavior for the CEP-dependent bound-state population at low and high intensities with a clear boundary, which we attribute to the transition from the multiphoton to the tunneling regime.
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