Asking for this directly could be very awkward, as you yourself point out. You could answer something like

Sure thing! It was a great gift by the way, it was exactly what I needed for my project. Keep it as long as you need!

This way, you can underline that the camera is now yours with a simple passing remark, without it being the focus of the discourse and calling too much attention. Also, since you take for granted that the camera is yours, the awkwardness if correcting you is now on her.

However, if she does state that she never intended to give you the camera as a present, you may want to minimise the "overall awkwardness" and say something like

Ah sorry, I must have misunderstood you. No problem, of course; I thought it was a gift because you told me I could keep it. Sorry again! Anyhow, what are the projects about?

That is, keep it short and simple, and straight to the point.

There's nothing inherently wrong or awkward in asking Jane directly whether the camera was a gift or not. However, as you yourself point out, it could be very awkward depending on the person you ask. If you don't feel comfortable asking Jane directly, you could answer something like

Sure thing! It was a great gift by the way, it was exactly what I needed for my project. Keep it as long as you need!

This way, you can underline that the camera is now yours with a simple passing remark, without it being the focus of the discourse and calling too much attention. Also, since you take for granted that the camera is yours, the awkwardness if correcting you is now on her.

However, if she does state that she never intended to give you the camera as a present, you may want to minimise the "overall awkwardness" and say something like

Ah sorry, I must have misunderstood you. No problem, of course; I thought it was a gift because you told me I could keep it. Sorry again! Anyhow, what are the projects about?

That is, keep it short and simple, and straight to the point.