Facebook is always changing things around so the specifics might vary a little, but this is how it could be done in April 2013.
From your news feed click on Add Photos/Video then Create Photo Album.
When you do this, a file selection box pops up to choose the file you want to add. You have to choose a file to upload, and you can even go back and delete it later, but this step is necessary because there is (apparently) no option to move an existing picture into a new album, or option to create an empty album.
Fill in the normal stuff like a name for the album, description, location, etc..
Here's the trick: Down next to the Post Photos button is a drop down thingie with options like Public, Friends, and Only Me, among others. Set this to Only Me. This prevents the new picture from appearing in your friends' news feeds or tickers before you can go and delete it.
Now go into your own profile/timeline and click Photos at the top.
Then click on Albums and click the album you want to move from.
Next not quite obvious thing: You need to click on the Edit button near the top of the page. Don't just click on the picture you want to move, because there's no Move button there!
In the Edit screen, if you mouse over the picture you want to move, there's a little down arrow pop-down control thingie that will appear the upper right corner of the picture. Move To Other Album is what you want.
Go into the photo album you created in the beginning and delete the placeholder picture you used to create the album. If you mouse over the picture, in the upper right corner a pencil icon will appear labeled Edit or Remove. Select that, then in the drop-down menu, Delete This Photo.
You should now have a new album with the pictures you moved into it, with all of the comments and likes preserved!
The last thing to do is change the permissions on the album from Only Me to whatever you want, like Friends. The permission control is at the top of the album view, right under the name of the album.
This is also where you could use Custom permissions to prevent some people (or lists of people) from seeing the album. Some special albums like iOS Photos allow you to set the permissions of each photo individually, but albums that you create you can only set the permissions for the entire album, which is why you might want to use this technique for moving pictures between albums.
That's it! If you've done this correctly there won't be any indication that you've done anything in your timeline or your friends' news feeds or tickers. And if you go back in your timeline to where you originally posted the picture, it will still be there, and clicking on it will open the new album as if it had always been there.