Unfortunately it is not possible to increase the size of an Amazon EBS root device storage volume while the Amazon EC2 instance is running - Eric Hammond has written a detailed (I'm inclined to say the 'canonical' ;) article about Resizing the Root Disk on a Running EBS Boot EC2 Instance:
As long as you are ok with a little down time on the EC2 instance (few
minutes), it is possible to change out the root EBS volume with a
larger copy, without needing to start a new instance.
If you properly prepare the steps he describes (I highly recommend to test them with a throw away EC2 instance first to get acquainted with the procedure), you should be able to finish the process with a few minutes downtime only indeed.
Good luck!
