Max current through an Arduino Mega? How much current can I put through the pins of an Arduino Mega at the same time?

The situation: I am building a 7x7x7 LED cube, and my plan was to multiplex in 7 layers of 49 pins, lighting each layer, one at a time. This would mean each pin on the Arduino Mega would only have to supply max. 1 LED at a time.
However, if all 49 LEDs in one layer were on, it would presumably draw 49*20mA (from the LED specs sheet), which is 980mA. This seems like quite a lot of current to put through an ATMega chip, so I was wondering if I could do it, either continuing like this, or using transistors to control the lighting of each layer, or what?
Any help or advice is appreciated!

EDIT: Using ATMega2560, not ATMega328 or alternate.
You're on the right track here. In the datasheet for the ATmega1280 I find this box:



What you want to look at is DC Current Vcc and GND Pins  ....  200.0 mA

This is the total limit of all the current going through the chip (because everything must come through Vcc), and it is the absolute maximum. Don't go near it. You need to boost this with some transistors.