Linking to YouTube while keeping you in your own LMS

Here's a scenario I'm sure we've all come across before:

On YouTube there is a video that you want to share with your audience for training purposes. However, you don't want to have a link to the video on the YouTube page because:

1. You want to keep people inside your eLeanring course and inside your LMS without going to a different window

2. You know that if you go to YouTube the likely hood of the user getting distracted by other videos/graphics/options are very high and it will interrupt your workflow

I suppose you could just screen capture/record the video with Captivate or or the option in SL to do so but there may be copywrite issues doing that which leads to all sorts of headaches.

So my question becomes:
Has anyone come up with a neat solution to be able to show the public content of YouTube housed somewhere in their own eLearning/LMS  that doesn't compromise the copywrite situation of the standard YouTube License agreement?

It would be neat to have another kind of window popup that displays the video in your eLearning which shows that it's a YouTube video (you can see the logo and the seek bar etc that clearly identify the source is YouTube) but it only displays that video in a window without the other YouTube elements to distract the user.

I'm open to suggestions and appreciate the support!
Thanks team!