Arduino Lamp Project - 2016

Hello all! I am a sorta newbie around here, posted a few Arduino-power related questions earlier, and now that the Holidays are over and I have time to devote to my project again, I'm back on board.

I have an Arduino project that I want to make into a Bluetooth controlled RGB lamp. I have the following pieces:

5V 2A Wall Power Outlet
Bluetooth adaptor
Small Breadboard
Adafruit 12 pixel RGB LED ring
Arduino Uno Rev 3 Board

These are the main parts of my project. the main problem that I have? is deciding what power supply will be sufficient enough for this project, as I already have to split the power supply for the adafruit using a DC Jack for ground and electricity. I tested my setup a few months back and while testing, I was careful to use separate power supplies for the Uno and the LED ring. Is there a way I could use a single power supply for both? I think last I inquired about this, I was told there would be just enough voltage but a problem with current or amps, not enough to fry anything, just a problem that would maybe cause neither to power up?

I don't want to make too many changes, as I planned this to be a USB power lamp in the beginning, and already had to make the change to wall socket. I know if I have to upgrade my power supply, it would probably have to be one of those really big ones, like the laptop chargers.

12 pixel neopixel ring should use about 720mA with all LEDs on max brightness.

A 5v/2A regulated wall adapter is suitable for powering it, but it's a little more complicated to get the power into the Arduino board, because you can't use the barrel jack (it goes through a regulator with 1.x volts of dropout, so you would only have 3.x volts on the Arduino's 5v line).

What I would do is as follows:

Connect Gnd of Arduino to Gnd of external supply, bluetooth, neopixel ring.

Connect +5v from external supply to neopixels.

Connect bluetooth supply to Arduino +5v.

Cut a USB cable up. Connect +5v and Gnd of the external power supply to red + black wires. To power Arduino, plug that into USB port. You could also connect the +5v of external power supply direct to the 5V pin of the Arduino, but you don't want to have that connected while the Arduino is plugged into USB. By using the USB cable to supply power, you ensure that you're only giving it 5v from that adapter when it's not plugged into USB.