[CSteefel]

Gandalf specifically says that Denethor apparently did not know the whole story of the Ring and he guessed that the scrolls of Isildur were read only by himself and Saruman:

And Boromir, there lies in Minas Tirith still, unread, I guess, by any save Saruman and myself since the kings failed, a scroll that Isildur made himself.

The verse about the Halfling and the discovery of Isildur's Bane was known to all, as was the fact that Isildur did not go straight to Arnor, but planted the White Tree in memory of his brother, but Gandalf says:

"'But in that time also he made this scroll', said Gandalf; 'and that is not remembered in Gondor, it would seem. For this scroll concerns the Ring..."

Early on, few knew of the existence of the Ring at all (with the exception of the Wise) and of those, most thought it was lost per the suggestions of Saruman. The focus in the first meeting with Denethor is on the possibility that Aragorn will be coming to claim the throne. However, in the second meeting upon the return of Faramir, Denethor seems to already know something of the meeting between Faramir and Frodo, and then states in referring to what Boromir would have done

He would have brought me a might gift.

That this is something like the Ring is explicitly acknowledged by Gandalf

'Comfort yourself!' said Gandalf. 'In no case would Boromir have brought it to you. He is dead, and died well; may he sleep in peace! Yet you deceive yoursef. He would have stretched out his hand to this thing, and taking it he would have fallen. He would have kept it for his own, and when he return you would not have known your son.'

This is followed by Denethor's response

'But I who was his father say that he would have brought it to me...I have in this matter more lore and wisdom than you deem.'

or even more explicitly

...would he have set this thing at hazard beyond all but a fool's hope, risking our uttter ruin, if the Enemy should recover what he lost.

which indicates pretty clearly that Denethor knew this was the Ring that Sauron had lost.

So it still is not clear whether he read the mind of Faramir, who figured out that this was the One Ring, or whether he knew it much earlier, in which case Gandalf seems to be mistaken about what Denethor knew early on. My supposition would be that he did not know the story of the Ring when he sent Boromir to Rivendell, but that he deduced it during later events, perhaps using the Palantir to read the mind of Faramir, since there is explicit reference to Denethor knowing much of what passed between Faramir and Frodo...