A search for fully reconstructed Lambda(b)(0) beauty baryons is performed using about 3 million Z decays collected with the DELPHI detector at LEP. The analysis relies on the combined use of the accurate tracking and of the hadron identification capabilities of DELPHI. A total of four events has been found, three in the Lambda(c)(+)pi(-) channel and one in the Lambda(c)(+)a(1)(-) channel over a small background. The Lambda(b)(0) beauty baryon mass is measured to be (5668 +/- 16 (stat.) +/-8 (syst.)) MeV/c(2).