What you want to do is fill the selection. You can do this using the Edit > Fill with FG Color Ctrl+, or by using the Bucket Tool.
If using the Bucket Tool, select the Bucket Tool Shift+B, and then Shift+click inside the selection. This will fill the selection with the current foreground color.
As MichaelSchumacher mentions in the comments, using the Bucket tool like this can cause some feathering around the edge of the selection.