We present an extinction map of the Large Magellanic Cloud (LMC), using 204,502 stars from the Two Micron All Sky Survey point source catalog. We first use the NICE method to determine the reddening distribution, \ehk and \ejh, which we compare to the HI distribution to find a near-infrared reddening law of $\ejh/\ehk=1.20\pm 0.04$. A visual extinction map ($\sim 6^\circ\times 6^\circ$) of the LMC is created using the NICER method; at 4 arcmin resolution, a mean value of $\av=0.38$ mag is found. We derive the LMC CO-to-H$_2$ conversion factor, $\x{LMC}$, independent of assumptions about the virialization of giant molecular clouds, by comparing the NICER extinction map with NANTEN $^{12}$CO observations. In regions where $\av>1$ mag and $^{12}$CO emission is $\ge$ 2 \counits, we measure $\x{LMC}=9.3\pm 0.4\times 10^{20} \xunits$. In the same regions, the LMC contains a total molecular mass of $(4.5\pm 0.2)\times 10^7 \msun$.

18 pages, 15 figures