Virgo is a kilometer-length interferometer for gravitationnal waves detection located near Pisa. Its first science run, VSR1, occured from May to October 2007. The aims of the calibration are to measure the detector sensitivity and to reconstruct the time series of the gravitationnal wave strain h(t). The absolute length calibration is based on an original non-linear reconstruction of the differential arm length variations in free swinging Michelson configurations. It uses the laser wavelength as length standard. This method is used to calibrate the frequency dependent response of the Virgo mirror actuators and derive the detector in-loop response and sensitivity within ∼ 5%. The principle of the strain reconstruction is highlighted and the h(t) systematic errors are estimated. A photon calibrator is used to check the sign of h(t). The reconstructed h(t) during VSR1 is valid from 10 Hz up to 10 kHz with systematic errors estimated to 6% in amplitude. The phase error is estimated to be 70 mrad below 1.9 kHz and 6 μs above. © 2010 IOP Publishing Ltd.