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The $λ$ = 2.06 $μ$m absorption band of CO$_2$ is widely used for the remote sensing of atmospheric carbon dioxide, making it relevant to many important top-down measurements of carbon flux. The forward models used in the retrieval algorithms employed in these measurements require increasingly accurate line intensity and line shape data from which absorption cross-sections can be computed. To overcome accuracy limitations of existing line lists, we used frequency-stabilized cavity ring-down spectroscopy to measure 39 transitions in the $^{12}$C$^{16}$O$_2$ absorption band. The line intensities were measured with an estimated relative combined standard uncertainty of $u_r$ = 0.08 %. We predict the $J$-dependence of the measured intensities using two theoretical modesl: a one-dimensional spectroscopic model with Herman-Wallis rotation-vibration corrections, and a line-by-line ab initio dipole moment surface model [Zak et al. JQSRT 2016;177:31-42]. For the second approach, we fit only a single factor to rescale the theoretical integrated band intensity to be consistent with the measured intensities. We find that the latter approach yields an equally adequate representation of the fitted $J$-dependent intensity data and provides the most physically general representation of the results. Our recommended value for the integrated band intensity equal to 7.183$\times$10$^{-21}$ cm molecule$^{-1}$ $\pm$ 6$\times$10$^{-24}$ cm molecule$^{-1}$ is based on the rescaled ab initio model and corresponds to a fitted scale factor of 1.0069 $\pm$ 0.0002. Comparisons of literature intensity values to our results reveal systematic deviations ranging from $-$1.16 % to +0.33 %.