How does one say a Private Mass - a Missa sine Populo? Well, of course, there is no such thing as a Private Mass. Any Mass, as Eric Mascall once explained, whether celebrated alone in a Saharan hermitage or in S Peter's Rome with cardinals galore and hundreds of thousands of the Faithful, is equally the Sacrifice of Calvary, the public sacrifice of Christ's Church, the great mystery of every place and age. Quite a thought as one stumbles up to the altar in an empty church on a freezing winter morning.
But we do use these slang terms as convenient shorthand for a Mass where the priest is assisted only by a server; or, even more reductively, by someone in the pews. (Incidentally, since the Motu proprio, laypeople who happen to know that there is such a Mass happening, are welcome to turn up, and that still doesn't stop it technically being sine populo - further proof of the point in my first paragraph: as well as of papa Ratzinger's creative cunning.) There is, however, an even more 'Private' Mass: one at which noone at all is present except the celebrating presbyter.
Until recently, such Masses were forbidden (not only in the C of E but also) by Vatican authority [General Instruction, 1969, para 211] nisi ex gravi necessitate. Such necessitas might have been the need to confect the Blessed Sacrament for somebody in articulo mortis. But the 1983 Codex Iuris Canonici modified this to nisi iusta et rationabili de causa - for a just and reasonable cause - and the semi-official Question Box of Msgr [now Bishop ... how these S Stephen's House liturgists do get around ...] Peter Elliott points out that Canon 904 'strongly recommends' daily celebration 'even if the faithful cannot be present'. The present Holy Father does not seem often to miss the chance of strongly commending daily celebration when addressing priests and seminarians. Now the General Instruction has been modified to bring it into line with the canon. And the old para 4, now para 19, has been modified to state that, in celebrating, the priest fulfills his principle role, so that he ought if possible to do so daily. This makes clear that even if there is another Mass that day in that church, or in another church to which the priest could get, at which he would be able to receive Communion but not to (con)celebrate, the laudable desire himself to celebrate would give him a just and reasonable cause to celebrate alone. If there were a concelebrated mass in which he could take part as a concelebrant, the matter might not be so clear.
I hope to do a post a little later about the ritual employed in doing this both in the Ordinary and in the Extraordinary Form of the Roman Rite.