My gut tells me that it would be for the remainder of THAT particular archery phase. The reason I think it specifies also that you cannot gain
archer, would prevent cards like that Orophin companion from just gaining archer again after losing it.
I kind of wondered why the card didn't just go the
Greenleaf/Gorn's Bow approach and make them "not add to the fellowship archery total", but doing it this way is slightly different as it would prevent you from further using cards like
Double Shot, etc.
I guess it is possible that it makes you lose it for the game, but I don't think so....Unless you were at site 9, that's a pretty substantial cost to just discard 1 minion.
Maybe bib could back this up as well