If the question concerns projective space specifically rather than projective geometry in a broader sense, then the answer would have to be Jean-Victor Poncelet (1788 – 1867). Desargues already introduced the notion of a point at infinity, but I believe Poncelet was the first to consider a LINE at infinity.

**Note** (added in 2023) I have researched the topic more closely recently in connection with a [publication on Leibniz](https://u.math.biu.ac.il/~katzmik/infinitesimals.html#23h).  It turns out that Desargues already had the notion of line at infinity.  There was recently a series of detailed articles on Desargues' work *Brouillon d'un Project*, by Anglade and Briend; for references see the article linked above.