What is the meaning of Milton's famous line, "That last infirmity of noble mind"?

The poem is an elegy to Milton's friend Edward King, who drowned while young. It is also a pastoral elegy, placed in a rural setting. In it, King is envisaged as having been a shepherd living in nature. Much of the poem is spoken by an imagined "uncouth swain," or shepherd friend of King's, lamenting his death.
In this passage, the shepherd is saying that the desire for fame draws men away from the pastoral life of pleasure, such as dallying with beautiful maidens like Amaryllis. Fame, instead wishes:

To scorn delights and live laborious days

The shepherd also says that

Fame is ...That last infirmity of noble mind

By this he means a noble person will put aside all other vices before abandoning the desire to be famous, because it can seem noble to pursue fame. The shepherd, however, questions whether it is worth the price to pursue fame rather than to enjoy the simple life. He goes on to say that lasting fame is not, anyway, bestowed on people in their lifetimes or by other people. It comes after death and is decided on by Zeus ("Jove"_):


all-judging Jove;
As he pronounces lastly on each deed

The shepherd argues, as a shepherd would, that fame is not worth pursing (it is an "infirmity") and is out of our hands.

Milton is translating a statement by the Roman historian Tacitus:
Etiam sapientibus cupido gloriae novissima exuitur. (The desire for glory clings even to the best men longer than any other passion.)Caius Cornelius Tacitus (A.D. c. 55 - c. 117)
Milton's translation of this profound thought is beautifully succinct in one graceful line of iambic pentameter. The statement implies that everyone has a desire to be important in one way or another, men and women alike. Wise men, however, recognize this desire for fame, glory, appreciation, admiration as a fault to be overcome, although it is the last of their faults for even the wisest men are able to overcome. Why is it a fault? It can lead to bad behavior as well as good. We should be inner directed, as Emerson says in his best essay "Self-Reliance," and not motivated by trying to please other people.