  Within Kirkwood theory, we study the translational diffusion coefficient of a
single polymer chain in dilute solution, and focus on the small difference
between the short--time Kirkwood value $D^{(K)}$ and the asymptotic long--time
value $D$. We calculate this correction term by highly accurate large--scale
Brownian Dynamics simulations, and show that it is in perfect agreement with
the rigorous variational result $D < D^{(K)}$, and with Fixman's Green--Kubo
formula, which is re--derived. This resolves the puzzle posed by earlier
numerical results (Rey {\em et al.}, Macromolecules 24, 4666 (1991)), which
rather seemed to indicate $D > D^{(K)}$; the older data are shown to have
insufficient statistical accuracy to resolve this question. We then discuss the
Green--Kubo integrand in some detail. This function behaves very differently
for pre--averaged vs. fluctuating hydrodynamics, as shown for the initial value
by analytical considerations corroborated by numerical results. We also present
further numerical data on the chain's statics and dynamics.
