We present a magnetoinfrared spectroscopy study on a newly identified three-dimensional (3D) Dirac semimetal ZrTe(5). We observe clear transitions between Landau levels and their further splitting under a magnetic field. Both the sequence of transitions and their field dependence follow quantitatively the relation expected for 3D massless Dirac fermions. The measurement also reveals an exceptionally low magnetic field needed to drive the compound into its quantum limit, demonstrating that ZrTe(5) is an extremely clean system and ideal platform for studying 3D Dirac fermions. The splitting of the Landau levels provides direct, bulk spectroscopic evidence that a relatively weak magnetic field can produce a sizable Zeeman effect on the 3D Dirac fermions, which lifts the spin degeneracy of Landau levels. Our analysis indicates that the compound evolves from a Dirac semimetal into a topological line-node semimetal under the current magnetic field configuration.