I am using "Jumping into C++" by Alex Allain. Seems very fine to me till now.I have also seen many others recommend this book. Everything is explained in way that can be understand by a newbie to programming.(i am one of those). You should find it fine. Hope this helps.
Hello Jakez. I own the book C++ Primer which JLBorges mentioned, albeit Fourth Edition. I have to admit though, if I didn't have a bit of previous programming experience, I wouldn't be able to use the book that well. So if you're going for the book (it is a pretty good one), make sure you have some previous experience. Also, here is a good resource for you:
I must add my favorite book here aswell. It is called Professional C++ and is written by Marc Gregoire who recieved Microsofts most valuble professional for his C++ knowledge.
It it so well written that it teaches you all the really powerfull and complex parts of C++ in an almost simple way. This book is the main reason that I went from knowing zero code to working as a System developer in 10 months without education :)