I know some others have given brief answers already, but this came up in the DS9 forum not long back, so I'll repost from that.
I suppose it depends on how "official" the contact is, or the degree of knowledge the Federation has. It's likely that the Federation worlds knew of a nation called Cardassia for decades if not centuries before official contact. At least, they knew that a world by that name was out there. Cardassia was, we've been told, part of the galactic community in some capacity as early as the 22nd century ("Enterprise" established that Cardassians had enough space-going capability to reach Organian haunts by then, and as others in this thread have noted, Iloja of Prim lived on Vulcan during that period). However, because Cardassia was presumably not a planet of note - just one of thousands of unremarkable worlds - it probably got entered into records without anyone in the Federation core sectors bothering with it or making official contact. A Vulcan ship trades with a Kressari ship, say, which previously traded with a Cardassian outpost - the Vulcans enter Cardassia into their cultural database as a minor, somewhat distant planet with which their trade partners interact. It was just a world out there somewhere beyond the Kalandra sector/Trill space. Or so I assume. I also assume Federation citizens and Cardassians no doubt met every now and then, but never officially, and their governments weren't concerned with each another. By the 2310s, however, it changes, because now the military expansion is beginning. Cardassia is no longer just an unremarkable world but a nascent local power.
We've had 2327 given as official first contact in several novels, including The Art of the Impossible (building on "Enter the Wolves", and TAotI indeed directly references the Legara conference from EtW, as does Well of Souls). Then there was 2321 given by a Maquis in The Brave and the Bold as the real first contact between Federation and Cardassian interests - civilian exchange on outer colonies, before the governments or militaries got involved, and Day of the Vipers hints at ship-based contact, fly-bys and skirmishes, for over a decade before that.
So, I assume that ships identified as Cardassian begin getting involved in fights with Federation-affiliated vessels along the border region around about the 2310s. Eventually, official first contact is made in 2327, after a few decades of unofficial "buzzings" and border incidents (and snooping around Bajor, according to Day of the Vipers). And then Cardassia blows any chance of good relations with its treachery at Legara, which in turn leads the Detapa Council and Obsidian Order to try and mend the relationship by having the UFP mediate the Betreka conflict the following year (TAotI).
Basically, while Cardassia had been know for centuries, it was only in the early 24th century that it became notable and invited official contact, and only then did it become anything other than an unremarkable border world among thousands of unremarkable border worlds.
Or so I assume.
Interesting stuff, Deranged; thanks for the info and the nudges towards those particular works.
