Yes, this is generally the case. If you fix a value like aperture (which is what you are doing when you set the camera to Av mode), one of the other exposure factors must change — and the only other options are shutter speed and ISO. If you are using automatic ISO, that may or may not change first, according to your specific camera's program line.

If you are using a fixed ISO value, the shutter speed is the only free variable. It will change up until your camera's limits — most shutters don't go faster than ¹⁄₄₀₀₀th of a second or so (some go to ¹⁄₈₀₀₀ or more, though), and particularly on point and shoot cameras, or mirrorless cameras (or DSLRs in live view), there may be a relatively short limit on the other side too. If you run up against one of these limits, obviously the camera can't automatically do anything about it, and what it does do will depend on the camera. (Underexpose? Refuse to take the picture? Depends.)

So, specifically, if you dial in +1 EC, shutter speed or ISO will increase by one stop, to make the exposure that much brighter than the meter reading would indicate. That means the shutter will stay open for twice as long — or the ISO sensitivity will double. (Or, theoretically, each could increase by half a stop, to get the same effect on exposure.)