We have performed high-resolution angle-resolved photoemission spectroscopy (ARPES) on trigonal tellurium consisting of helical chains in the crystal. Through the band-structure mapping in the three-dimensional Brillouin zone, we found a definitive evidence for the band splitting originating from the chiral nature of crystal. A direct comparison of the band dispersion between the ARPES results and the first-principles band-structure calculations suggests the presence of Weyl nodes and tiny spin-polarized hole pockets around the H point. The present result opens a pathway toward studying the interplay among crystal symmetry, band structure, and exotic physical properties in chiral crystals.