The Supernova Remnant (SNR) IC 443 is an intermediate-age remnant well known for its radio, optical, X-ray and gamma-ray energy emissions. In this Letter we study the gamma-ray emission above 100 MeV from IC 443 as obtained by the AGILE satellite. A distinct pattern of diffuse emission in the energy range 100 MeV-3 GeV is detected across the SNR with its prominent maximum (source A) localized in the Northeastern shell with a flux F = (47 pm 10) 10^{-8} photons cm^{-2} s^{-1} above 100 MeV. This location is the site of the strongest shock interaction between the SNR blast wave and the dense circumstellar medium. Source A is not coincident with the TeV source located 0.4 degree away and associated with a dense molecular cloud complex in the SNR central region. From our observations, and from the lack of detectable diffuse TeV emission from its Northeastern rim, we demonstrate that electrons cannot be the main emitters of gamma-rays in the range 0.1-10 GeV at the site of the strongest SNR shock. The intensity, spectral characteristics, and location of the most prominent gamma-ray emission together with the absence of co-spatial detectable TeV emission are consistent only with a hadronic model of cosmic-ray acceleration in the SNR. A high-density molecular cloud (cloud E) provides a remarkable target for nucleonic interactions of accelerated hadrons: our results show enhanced gamma-ray production near the molecular cloud/shocked shell interaction site. IC 443 provides the first unambiguous evidence of cosmic-ray acceleration by SNRs.