No, say all the articles on this topic, we can't build a dinosaur from dino-DNA extracted from a mosquito trapped in amber. In fact, we can't build a true dinosaur at all. But paleontologist Jack Horner has been working on a project that's still pretty awesome:
But even though we didn't find DNA in an extinct dinosaur, I decided to see if we could retro-engineer a living dinosaur — all birds are living dinosaurs — and make it look like an extinct dinosaur.
The fossil, a tiny neck bone found in the southern U.K., is a mere quarter-inch (7.1 millimeters) in length. It belongs to an adult dinosaur that lived in the Cretaceous period 145 million to 100 million years ago, reported University of Portsmouth paleozoologist Darren Naish in the August issue of the journal Cretaceous Research.
The bone was found by an amateur fossil hunter who didn't realize what he had... he kept the bone in his bedside table for two years before showing it to an expert. More on the find here and here.