The "artifice of eternity" is the beautiful mechanical bird the speaker would like to become. In the phrase "artifice if eternity", the word "artifice" means an artificial mechanism.
The speaker is an old man who feels that the "country" of nature is no home for him anymore. To be natural—to be born—means he will soon die because he is old. As he states, he no longer wants to feel a part of:
Whatever is begotten, born, and dies
Therefore, he dreams of being made into something artificial—something crafted by an artisan, or, in other words, a work of artifice that will never die. He says he desires that his soul be separated from his body and gathered into the eternal. For him, this eternity means becoming, as he puts it in stanza 4, a work of art pounded out of gold that will live forever:
such a form as Grecian goldsmiths make
Of hammered gold and gold enamelling