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 µm photometry. The Eye is detected with high significance at both 24 and 70 µm and, when including a flux limit at 3.5 mm, we estimate an infrared luminosity of L IR = 8.3 +4.7 −4.4 × 10 11 L ⊙ 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 µm, with equivalent widths near the maximum values observed in star-forming galaxies at any redshift. The L P AH -to-L IR ratio lies close to the relation measured in local starbursts. Therefore, L P AH 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 µm 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 × 10 −4 , are both in agreement with measurements in local starbursts and ULIRGs, suggesting that this line may serve as a good proxy for L P AH or L IR at z > 3 with the James Webb Space Telescope.