Copy Worksheet Between Books and Retain Formatting

I have the following code that copies a specific worksheet from one workbook,
to another. Problem is, that the format of the worksheet is lost during the
copy. Is there a way to make sure that, when copied, the worksheet I have
just pasted looks exactly like the original?

Thanks for any help! Patk

Sub SheetCopyBetweenFiles(sourcefile, destfile, sheetnametocopy, aftersheet)

Dim srcbook, destbook As Workbook
Dim copysheet, newsheet As Worksheet

Workbooks(sourcefile).Activate
Set srcbook = ActiveWorkbook
Set copysheet = srcbook.Worksheets(sheetnametocopy)
Set destbook = Workbooks(destfile)

With destbook
.Worksheets.Add after:=.Worksheets(.Worksheets.Count)
.ActiveSheet.Name = sheetnametocopy
copysheet.Cells.Copy .Worksheets(.Worksheets.Count).Cells

End With
Set srcbook = Nothing
Set copysheet = Nothing
Set destbook = Nothing
End Sub