One way to do this is to use the RichTextBox.SaveFile method.

private void button1_Click( object sender, EventArgs e )
{
  // get a file name from the user
  SaveFileDialog saveFile1 = new SaveFileDialog();
  saveFile1.DefaultExt = "*.rtf";
  saveFile1.Filter = "RTF Files|*.rtf";
  if ( saveFile1.ShowDialog() == DialogResult.OK )
    richTextBox1.SaveFile( saveFile1.FileName, RichTextBoxStreamType.RichText )
}

Contributed from George Shepherd's Windows Forms FAQ