[Boromir88]

What's interesting about Isildur's Bane, is that it doesn't seem like the Gondorians at this time understood it was The One Ring. Legate gives Faramir's answer to Frodo and it always looked to me like Faramir attributed Isildur's Bane to the Orc arrows that killed him...then later upon finding out Frodo had Sauron's Ring he surmises that this was Isildur's bane. However, from the quote that Legate gives, it certainly appears that Faramir at first thought Isildur's Bane was the Orc arrows that killed him, then through some consideration he reaches the conclusion that it was actually the Ring.

What Boromir tells the Council about possibly what Denethor knew:

Seek for the Sword that was broken:
In Imladris it dwells;
There shall be counsels taken
Stronger than Morgul-spells.
There shall be shown a great token
That Doom is near at hand,
For Isildur's Bane shall waken,
And the Halfling forth shall stand.

Of these words we could understand little, and we spoke to our father, Denethor, Lord of Minas Tirith, wise in the lore of Gondor. This only would he say, that Imladris was of old the name among the Elves of a far northern dale, where Elrond the Halfelven dwelt, greatest of lore-masters.~The Council of Elrond

Denethor only revealed the 'riddle' about where Imladris was at, that doesn't mean he didn't know more, that's just all he chose to tell his sons. Denethor I would think had actually guessed more considering him reluctantly letting Boromir go on this trek to Rivendell:

Therefore my brother, seeing how desperate was our need, was eager to heed the dream and seek for Imladris; but since the way was full of doubt and danger, I took the journey upon myself. Loth was my father to give me leave,...~ibid

Denethor is very hesistant about Boromir going, but since Boromir (according to Faramir would have it no other way) Denethor finally gave in. I think it's safe to say that Denethor had some worries about this riddle and this trek to Rivendell as he was very hesistant about letting Boromir go.

Basically I'm saying that Denethor probably knew more than what he had told his sons. Or at least this riddle of 'Isildur's Bane' began to trouble him and he went further to investigate. Denethor definitely had the access to the records in Gondor, he even had access to records that no one else did, and only the Steward could view...for example all the lore in Gondor concerning the palantiri. Where if I remember correctly (from Unfinished Tales; The palantiri) only the Steward and the Steward's heir had access to these. Denethor definitely had access to all the 'secret' accounts hidden from the rest of the Gondorian public. Boromir says he's Gondor's greatest Loremaster, and he and Faramir actually went to their father to try to decipher this riddle. Denethor is no chump and no oblivious idiot when it comes to matters concerning his Kingdom. Isildur's Bane would definitely be a matter concerning his Kingdom, and I think Denethor would be aware of this...he may have had some guesses and thoughts before, but I'm sure he ended up figuring out the answers...some descriptions of Denethor:

"...nor had he [Sauron] any servant whose mental powers were superior to Saruman's or even Denethor's."~Unfinished Tales; The Palantiri

In this way [viewing the palantir] Denethor gained his great knowledge of things that passed in his realm, and far beyond his borders, at which men marvelled; but he bough the knowledge dearly, being aged before his time by his contest with the will of Sauron.'~Appendix A: Gondor and the Heirs of Anarion; The Stewards

When it came to matters concering with Gondor Denethor was no oblivious fool. If it something that concerned Gondor, he was going to eventually find out about it. I don't know whether Denethor knew at first what Isildur's Bane really was, he may of had some guesses, but I don't think there's doubt he soon found out about it being the Ring. As Denethor had access to all these records, and when it concerned Gondor he wanted as much information as he could get.