I'm not 100% sure on the browser compatibility of this, but you can have two div's - an outer div and a inner div. The inner div will have all your content. Your css could then look like this:
#outer {
width: 200px;
height: 200px;
overflow:hidden;
}
#inner {
height: 200px;
width: 225px;
overflow: scroll;
}
That is, the inner block would be wide enough to contain a scrollbar, but have it hidden from sight by the containing div. This worked for me in webkit. You might have to fiddle with the widths to make sure text doesn't get cut off.
That said, I would carefully think about WHY you're trying to do this. This could be a huge usability issue for your users, as they will not have any indication that there is more content within the div.