We demonstrate a new approach for probing the atmospheres of selected chemically peculiar magnetic stars. Using the Cerro Tololo Inter-American Observatory (CTIO) 1.0 and 1.5m telescopes simultaneously on two nights in 1991, we obtained optical and infrared high-speed photometry of the rapidly oscillating Ap (roAp) star HR 3831 (=HD 83368), which pulsates with a period of ~11.8 minutes. Oscillation amplitudes (or upper limits) in each of eight bandpasses were measured by Fourier analysis, revealing the very steep decline in amplitude with increasing wavelength characteristic of roAp stars. Matthews and coworkers have shown that this can be explained by the wavelength dependence of limb darkening and its filtering effect on the integrated amplitude of an (l,m)=(1,0) nonradial pulsation mode. Since the eigenfrequency spectrum of HR 3831 is dominated by just such a dipole mode, it is possible to infer limb-darkening coefficients beta(lambda) for its atmosphere from our amplitude measurements. These coefficients are used to derive a source function through the Eddington-Barber relation, which leads to a relation for temperature versus optical depth in the stellar photosphere. Our results show that the T-Tau(5000) curve is much steeper for HR 3831 than for the Sun (at least at the rotational phase when the star's magnetic pole dominates the visible hemisphere). We also calculate the gradient dT/dtau as a function of wavelength to compare with the curve expected for H- continuous opacity. This type of diagnostic may become a useful tool for checking the predictions of diffusion theory.