  The potential of the TESLA linear e+e- collider to determine the strong
coupling constant, alpha_S, at 500 GeV is investigated. Experimental
complications due to background from W- and Z-pairs, top-production, initial
state photon radiation and from beamstrahlung are considered. The hadronic
event selection procedures used by the experiments at LEP II are reviewed for
the applicability at TESLA. An estimate of the various error contributions to
the total uncertainty of an $\alpha_S$ determination is presented. It confirms
that hadronisation effects are diminished while the uncertainty from the choice
of the renormalisation scale will dominate. Fits of the ln(R)-matched second
order (O(alpha_S^2)) and resummed calculation (NLLA) to six observables are
used to estimate the error contributions. This yields the expectation of the
precision for alpha_S(500 GeV) of +-0.0025.
