Can I safely connect my original Arduino Uno to a 19V adapter?
According to the page you linked:


  The board can operate on an external supply of 6 to 20 volts. If supplied with less than 7V, however, the 5V pin may supply less than five volts and the board may be unstable. If using more than 12V, the voltage regulator may overheat and damage the board. The recommended range is 7 to 12 volts.


So, yes. You can use the 19V external power supply to power the board. However, be aware that the voltage regulator can overheat over time and damage the board.