<p>Is there a built-in editor for a multi-line string in a <code>PropertyGrid</code>.</p>

No, you will need to create what's called a modal UI type editor. You'll need to create a class that inherits from UITypeEditor. This is basically a form that gets shown when you click on the ellipsis button on the right side of the property you are editing.
The only drawback I found, was that I needed to decorate the specific string property with a specific attribute. It's been a while since I had to do that. I got this information from a book by Chris Sells called "Windows Forms Programming in C#"
There's a commercial propertygrid called
Smart PropertyGrid.NET
by VisualHint.