Copy From LastRow to LastRow+7

Heads up... when working with Row variables use Long not Integer given row numbers exceed Integer boundaries (also in 32 Bit VBA Integers get converted to Longs under the hood anyway)

If you want code that will work on all versions best to use Rows.Count than 65536 - this is because in XL2007 the available rows increased (1m+) and if you upgrade at a later date the use of 65536 is obviously open to error.

Sub AddNewProject()
Dim LastRow As Long
LastRow = Cells(Rows.Count,"A").End(xlUp).Row
Rows(LastRow).Resize(8).Copy
End Sub

Or if you prefer to avoid Resize

Rows(LastRow & ":" & LastRow + 7).Copy