Need multiple date stamps in ColA

I'm appending data to rows (ColB:ColL) in one w/s from another. Each time rows are appended I want a date stamp in ColA to be entered automatically. When multiple rows are appended the following code only puts the date in the first appended row.

Private Sub Worksheet_Change(ByVal Target As Range)
If Target.Column <> 2 Or Target.Row = 1 Then Exit Sub
Cells(Target.Row, 1) = Date
End Sub

Thanks,
roothog

Maybe try something like this...

Private Sub Worksheet_Change(ByVal Target As Range)
    If Target.Column <> 2 Or Target.Row = 1 Then Exit Sub
    Intersect(Range("A:A"), Target.EntireRow) = Date
End Sub

That worked AlphaFrog, in putting the time stamp in each row copied. I didn't think it would make a difference and should've noted, but the data appended most often has blank rows at the bottom of the copy/pastespecial (19 rows copied each time filled top down). So, if the row is blank (always at the bottom) there needs to be no date stamp.

Private Sub Worksheet_Change(ByVal Target As Range)
    Dim cell As Range
    If Intersect(Range("B2:B" & Rows.Count), Target) Is Nothing Then Exit Sub
    On Error GoTo ReEnable
    Application.EnableEvents = False
    For Each cell In Intersect(Range("B2:B" & Rows.Count), Target)
        If cell.Value <> "" Then cell.Offset(, -1).Value = Date
    Next cell
ReEnable:
    Application.EnableEvents = True
    If Err.Number <> 0 Then MsgBox Err.Number & " " & Err.Description, vbCritical, _
                                   "Error in Worksheet_Change Procedure"
End Sub

a b s o l u t e l y perfect! thanks AlphaFrog for you time and knowledge for solving my problem!

roothog