You can't draw 1000 mA from an Arduino pin. 40 mA is the absolute max. 20 or 30 mA would be a good limit. You can use an Arduino pin to switch higher currents using a transistor. A Google search for 'Arduino transistor' will turn up examples and tutorials.
You can't draw 1000 mA from the Arduino 5V regulator. That's the regulator's limit and you need some power to run the Arduino.
You can't draw 1000 mA from the USB cable. They are limited to 500 mA.
If the resistance of the wire is much more than 5 Ohms you will need a voltage higher than 5V to push 1A through it.
If the resistance of the wire is much less than the voltage you are using you will need some way to limit the current so you don't melt the wire. A series resitor would do it but will need to be able to handle the power dissipation.