Remove the battery and verify the the LED on the AC adapter is on before you plug in the notebook end.  Then try powering up.  If the system will power up, the battery is the issue.  If it won't, the mainboard is more likely the faulty part.  
Hello there,   I have for about 10 years an Adamo 13 laptop. That's a great piece of hardware and probably the most beautiful laptop ever made. Recently, the laptop does not simly power up anymore when pushing the power button. This laptop is  always used with power plugged in. When plugged, I still see the light on the top back of the laptop, it simply does not power up. Could it possible the battery is simply dead and, even though the power plug is plugged in, can't simply power up becasue the battery is 'required'? Something else?   Thanks in advance for your help! B/R A.B.
Dell Adamo 13 (non-xps): not powering up anymore