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We report the discovery of TeV gamma-ray emission from the Type Ia supernova remnant (SNR) G120.1+1.4, known as Tycho's SNR. Observations performed in the period 2008-2010 with the VERITAS ground-based gamma-ray observatory reveal weak emission coming from the direction of the remnant, compatible with a point source located at 00(h)25(m)27(s).0, +64 degrees 10'50 '' (J2000). The TeV photon spectrum measured by VERITAS can be described with a power law dN/dE = C(E/3.42 TeV)(-Gamma) with Gamma = 1.95 +/- 0.51(stat) +/- 0.30(sys) and C = (1.55 +/- 0.43(stat) +/- 0.47(sys)) x 10(-14) cm(-2) s(-1) TeV-1. The integral flux above 1 TeV corresponds to similar to 0.9% of the steady Crab Nebula emission above the same energy, making it one of the weakest sources yet detected in TeV gamma rays. We present both leptonic and hadronic models that can describe the data. The lowest magnetic field allowed in these models is similar to 80 mu G, which may be interpreted as evidence for magnetic field amplification.