WiFi needs current. Get a nice, hard +5 power supply and put a switch in the USB +5 (red) wire to prevent back feeding if you turn the project on first.
The maximum setting on the adapter is 12 volts, though for fear of frying the board, I never set it that high (I'm not a complete idiot, only a part-time one); the other settings are 3, 4.5, 5, 6, 7.5, and 9 volts. I should have stated this in the first place, my mistake.
According to the UNO's tech specs, the recommended input is 7-12V.
Use the lowest functional voltage. The regulator has to dissipate all the heat in lowering the voltage to +5. +12 puts 7V across the regulator. With a maximum current of 200mA and P = E x I, E is 7 x .2 = 1.4W to be dissipated in a SOT package.
At 7V input the differential is 2V x .2 = .4W, a LOT less.
These types of linear reg will often fail shorted, applying the full unregulated voltage to the card. Lower operating voltage and temp reduce the risk of failure. Electronic life is derated by 1/2 for every 10 deg F. Running at 180F greatly reduces the life vs 90F
That is only for the input to the Vin pin or the power jack.
That then goes through a regulator which cuts it down to 5V, and that powers the chip.
That often leads beginners to think you can put that voltage range on the pins of the actual microprocessor chip. No voltage on the chip should exceed 0.7V greater than the outage powering the chip. This gives a little wriggle room if the internal regulator is not quite spot on 5V, which if you measure it you will find out.
Another thing I forgot to mention: when I was powering the board via USB, the other end of the USB cable was connected to the computer. I've also tried using the charger (5.2 V) that came with my iPad to power the board via the USB cable (the one for the Arduino that is, not the one for the iPad).