I've re-read it and it's definitely the part of Voldemort's soul that had attached to Harry.
The whole last quarter of the book has to be re-read, I think, just to get an actual grip of what has happened. I found I read it too quickly due to intrigue, excitement and speed of plot the first time round.
I'd like to think Voldemort had a similar conversation with Dumbledore in some abstract place of his inadvertant choosing and that infant part of his soul helped him in some way.