Infrared extinction maps and submillimeter dust continuum maps are powerful probes of the density structure in the envelope of star-forming cores. We make a direct comparison between infrared and submillimeter dust continuum observations of the low-mass Class 0 core, B335, to constrain the ratio of submillimeter to infrared opacity (\kaprat) and the submillimeter opacity power-law index ($κ ∝ λ^{-β}$). Using the average value of theoretical dust opacity models at 2.2 \micron, we constrain the dust opacity at 850 and 450 \micron . Using new dust continuum models based upon the broken power-law density structure derived from interferometric observations of B335 and the infall model derived from molecular line observations of B335, we find that the opacity ratios are $\frac{\kappa_{850}}{κ_{2.2}} = (3.21 - 4.80)^{+0.44}_{-0.30} \times 10^{-4}$ and $\frac{\kappa_{450}}{κ_{2.2}} = (12.8 - 24.8)^{+2.4}_{-1.3} \times 10^{-4}$ with a submillimeter opacity power-law index of $β_{smm} = (2.18 - 2.58)^{+0.30}_{-0.30}$. The range of quoted values are determined from the uncertainty in the physical model for B335. For an average 2.2 \micron\ opacity of $3800 ± 700$ cm$^2$g$^{-1}$, we find a dust opacity at 850 and 450 \micron\ of $κ_{850} = (1.18 - 1.77)^{+0.36}_{-0.24}$ and $κ_{450} = (4.72 - 9.13)^{+1.9}_{-0.98}$ cm$^2$g$^{-1}$ of dust. These opacities are from $(0.65 - 0.97) κ^{\rm{OH}5}_{850}$ of the widely used theoretical opacities of Ossenkopf and Henning for coagulated ice grains with thin mantles at 850\micron. ; Comment: 28 pages, 7 figure. Accepted for publication in ApJ