Consider:
Private Sub Worksheet_Change(ByVal Target As Range)
If Target.Address(0, 0) = "B1" Then
Application.EnableEvents = False
Range("A1").Value = 2 * Range("B1").Value
Application.EnableEvents = True
End If
End Sub
EDIT#1:
If this fails to work, but does not produce an error message, you may need to re-enable Events . Put this in a standard module and run it:
Sub EventsOn()
Application.EnableEvents = True
End Sub