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At first install davfs2 support by typing:
$ sudo apt-get install davfs2
Create a file (this will be a launcher icon) on yout desktop by typing in console. You can use any editor you like, my fave is vi so I will use that:
$ vi ~/Desktop/Box.desktop
Paste this thing into the Box.desktop file:
#!/usr/bin/env xdg-open
[Desktop Entry]
Encoding=UTF-8
Version=1.0
Type=Link
Icon[en_US]=gnome-panel-launcher
Name[en_US]=Box
URL=dav://www.box.net/dav
Name=Box
Icon=/usr/share/icons/Human/48x48/places/folder-remote.png
Now just doubleclick on the desktop icon. You will be prompted for login and password. You can make ubuntu remember your password by selecting remember password forever. You can also edit file /etc/davfs2/secrets to store your password and login there.