The high-frequency-peaked BL Lacertae object 1ES 0806+524, at redshift z = 0.138, was observed in the very high energy (VHE) gamma-ray regime by VERITAS between 2006 November and 2008 April. These data encompass the two- and three-telescope commissioning phases, as well as observations with the full four-telescope array. 1ES 0806+524 is detected with a statistical significance of 6.3 standard deviations from 245 excess events. Little or no measurable variability on monthly timescales is found. The photon spectrum for the period 2007 November to 2008 April can be characterized by a power law with photon index 3.6 +/- 1.0(stat) +/- 0.3(sys) between similar to 300 GeV and similar to 700 GeV. The integral flux above 300 GeV is ( 2.2 +/- 0.5(stat) +/- 0.4(sys)) x 10(-12)cm(-2)s(-1) which corresponds to 1.8% of the Crab Nebula flux. Non-contemporaneous multiwavelength observations are combined with the VHE data to produce a broadband spectral energy distribution that can be reasonably described using a synchrotron-self-Compton model.