A detailed analysis of a large collection of electronic spectra from three observatories, together with radial velocities published earlier, were used to derive a new ephemeris and improved orbital elements for the e Per binary. Observations covering a time interval of about 37 000 days (101.3 years) can be reconciled with a constant orbital period of 14(d).06916 +/- 0(d).00004. The high orbital eccentricity of 0.555 +/- 0.009 was also confirmed. New spectral observations confirm that there is a periodic variation of the systemic velocity. Together with new evidence from astrometric observations ( also analyzed here), they confirm the existence of a third body in the system with an orbital period of about 9600 days (26.3 years), rather than 4156 days, as reported earlier. Application of the disentangling technique to the Ha spectra with good S/N ratios did not allow detection of spectral lines of either the secondary or tertiary components. For plausible inclinations between 30 degrees and 90 degrees, the observed mass function implies a mass of the secondary M-2 = 0.85-1.77 M-circle dot, if a primary mass is adopted of 13.5 +/- 2.0 M-circle dot. Attempts to detect the third body via interferometric observations should continue in spite of this first negative result.