How to create a WPF Window without a border that can be resized via a grip only?

Question

If you set ResizeMode="CanResizeWithGrip" on a WPF Window then a resize grip is shown in the lower right corner, as below:

If you set WindowStyle="None" as well the title bar disappears but the grey bevelled edge remains until you set ResizeMode="NoResize". Unfortunately, with this combination of properties set, the resize grip also disappears.

I have overridden the Window's ControlTemplate via a custom Style. I want to specify the border of the window myself, and I don't need users to be able to resize the window from all four sides, but I do need a resize grip.

Can someone detail a simple way to meet all of these criteria?

1. Do not have a border on the Window apart from the one I specify myself in a ControlTemplate.
2. Do have a working resize grip in the lower right corner.
3. Do not have a title bar.