Believe it or not, there is an amplifier inside that small antenna. The antenna you linked to is unusual in that it accepts anything from 2V5 to 12V. Most don't go above 5V. Some older active systems used 12V, but today many have gone below 5V to 3V3.
On the datasheet for the GPS module, the description for pin 10 says that it provides 5V for an active antenna. So yes, you connect the center pin of the antenna's SMA connector to pin 10. As blueskull said, some GPS modules require external circuitry for the antenna, but this Pletronics unit doesn't.
So your 'Black Box' is nothing more than a piece of wire. Do NOT connect the 5V lead shown on your drawing to the antenna. That connection is handled inside the module. Of course, since that wire also carries the 1.574 GHz GPS signal, some care and attention is required to ensure that the signal isn't degraded.