  In this paper, an autonomous Choquard equation with the upper critical
exponent is considered. By using the Poho\v{z}aev constraint method, the
subcritical approximation method and the compactness lemma of Strauss, a
groundstate solution in $H^1(\mathbb{R}^N)$ which is positive and radially
symmetric is obtained. The result here extends and complements the earlier
theorems.
