We report the results of a Spitzer infrared study of the Cosmic Eye, a strongly lensed, L*_UV Lyman Break Galaxy (LBG) at z=3.074. We obtained Spitzer IRS spectroscopy as well as MIPS 24 and 70 micron photometry. The Eye is detected with high significance at both 24 and 70 microns and, when including a flux limit at 3.5 mm, we estimate an infrared luminosity of L_IR = 8.3 (+4.7-4.4) x10^11 L_sun assuming a magnification of 28+-3. This L_IR is eight times lower than that predicted from the rest-frame UV properties assuming a Calzetti reddening law. This has also been observed in other young LBGs, and indicates that the dust reddening law may be steeper in these galaxies. The mid-IR spectrum shows strong PAH emission at 6.2 and 7.7 microns, with equivalent widths near the maximum values observed in star-forming galaxies at any redshift. The L_PAH-to-L_IR ratio lies close to the relation measured in local starbursts. Therefore, L_PAH or L_MIR may be used to estimate L_IR and thus, star formation rate, of LBGs, whose fluxes at longer wavelengths are typically below current confusion limits. We also report the highest redshift detection of the 3.3 micron PAH emission feature. The PAH ratio, L_6.2/L_3.3=5.1+- 2.7, and the PAH-to-L_IR ratio, L_3.3/L_IR = 8.5 +- 4.7 x10^-4, are both in agreement with measurements in local starbursts and ULIRGs, suggesting that this line may serve as a good proxy for L_PAH or L_IR at z > 3 with the James Webb Space Telescope.
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