Glich in youtube brocast your self

1. Open a song you want to enjoy in the YouTube Music app.
2. You will see the option to select Song or Video mode at the top of the screen. Tap Video to turn on video mode. Tap Song to only hear the album version of the song.
3. If you want to stop playing videos and go back to a listening only experience, tap Song.

Is it possible that you have not download the whole music actually. How about giving this Leawo YouTube Downloader a try?

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Hope it helps. :) Watch online videos offline freely with Leawo YouTube Downloader

If it stops and go, it is due to the speed of your Internet connection. You might want to start the song and pause it for a little bit, and play. This way you gives your pc/laptop enough time to download the song so that you won't have the stopping issue. Good luck

Youtube downloads the song in the background as you listen to it. If this process, called buffering, happens too slowly, then youtube will pause.
Make sure you aren't downloading anything in the background, otherwise you'll need to load the page, pause the video, and let it sit for a few moments before playing. Getting a faster connection can solve this problem.
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downloading videos from youtube ,please read this tutorial:
<a href=http://www.aneesoft.com/tutorials/free-youtube-downloader/download-youtube-video-online-free.html>How to download online YouTube videos for free</a>

Try youtube to mp3 downloader.
http://onlinevideodownloader.net/how-to-download-songs-from-youtube.html
you can easily download youtube as audio file

try powering down the phone and powering it back up it might be a glich in the firm where

Just reset the Zune by holding the left button and the up portion of the center control continuously until the screen goes blank. Release both and the unit will reboot.

Here's a Youtube showing the process: http://www.youtube.com/watch?v=8BqqlDwtYbQ

If this doesn't work, contact MS to get a warranty replacement.

Assuming it does work, be sure to check that you have the latest version of the Zune SW as new units are usually several versions old and updating your unit will make it more reliable in my experience.