procedure TForm1.ListViewCheckStateChanging(Sender: TObject; Item: TListItem; ToBeChecked: Boolean;
  var AllowChange: Boolean);
var
  I: Integer;
begin
  // detach this event handler
  ListView1.OnCheckStateChanging := nil;
  try
    // iterate all the items unless we find a checked one
    for I := 0 to ListView1.Items.Count - 1 do
      // we have found a checked item, so...
      if ListView1.Items[I].Checked then
      begin
        // allow changing only when index of the item to be changed
        // differs from the currently iterated one; that is here to
        // prevent unchecking an already checked item
        AllowChange := Item.Index <> I;
        // if the change is allowed; it means this iterated item is
        // checked and it's not the one which is going to be changed,
        // uncheck it
        if AllowChange then
          ListView1.Items[I].Checked := False;
        // and break the loop since we don't have more than 1 checked
        // item
        Break;
      end;
  finally
    // finally re-attach the event handler
    ListView1.OnCheckStateChanging := ListViewCheckStateChanging;
  end;
end;