The high crystallinity of many inorganic materials allows their band structures to be determined through angle-resolved photoemission spectroscopy (ARPES). Similar studies of conjugated organic molecules of interest in optoelectronics are often hampered by difficulties in growing well-ordered and well-oriented crystals or films. We have grown crystalline films of uniaxially oriented sexiphenyl molecules and obtained ARPES data. Supported by density-functional calculations, we show that, in the direction parallel to the principal molecular axis, a quasi-one-dimensional band structure of a system of well-defined finite size develops out of individual molecular orbitals. In contrast, perpendicular to the molecules, the band structure reflects the periodicity of the molecular crystal, and continuous bands with a large dispersion were observed.