Yes that is correct. There can only be one colour profile at a time in a single file.
If I am not wrong, you will either see a colour shift when you drag the aRGB layer over to the sRGB layer, or else if you turned on the colour management warnings, it will warn you that the colour profile does not match. If you ignore the mismatch and go ahead to save the file, it will be saved as sRGB. Just remember that your aRGB layer will be interpreted as sRGB and there may be a drastic colour difference between your original file colour of the aRGB layer and the colour in the file with the combined layers.
To avoid getting a drastic colour shift, first convert your aRGB to sRGB profile (can't remember exact steps cos' my office computer doesn't have photoshop), and then drag the layer over to the sRGB layer.
You can only have one colour profile per file, so you have to make a choice between preserving the look of the colour, or the actual colour numbers. This is because the colour spaces are of a different size and shape. So no, you can't preserve the colour profile if you mean you want to preserve both the colour from the original pic and the colour values assigned to it.
ie if you have a pic of a rose, and the red is for example 230R 0G 0B in aRGB, you need to decide if you want it to look the same colour when converted to sRGB, or if you want the colour values 230R 0G 0B to be retained (ie colour may not be the same). Keep in mind that 230R 0G 0B in sRGB may be a lighter shade of red or even pink.
So normally, you would want to preserve the colour rendition and not the colour values. So I would probably want to allow photoshop to map the equivalent values from aRGB to sRGB so that the colours are reproduced as faithfully as possible.
You can test this out by selecting a part of the picture with flat colour and using the eye dropper tool to see the colour numbers before and after you move the layer. In most cases, the numbers will shift in order to keep the colour constant.
You realise there are actually 3 profiles? Source, destination and working. You don't have to convert from one profile to another first because that is automatically done for you when you open the file, if you choose not to convert, your colours will be wrong. Since your working profile is set to Adobe RGB, whatever file will be (has to be) converted to A-RGB to preserve the colour accuracy. So when you're cutting and pasting across files, the colours will be in the A-RGB colour space.
Then there's your output profile. After you work, you want to save the file, then there will be another conversion. If you're going to print it within Photoshop environment again next time, best to save as A-RGB, because your working space is A-RGB. Otherwise once you convert, you will not reap the benefit of the extra gamut of the A-RGB profile because there will definitely be some colours which will not be mapped.
At the end of the day, you have to decide what is your output going to be and decide from there. If you're going to use A-RGB, then you should shoot A-RGB also to minimize the need for conversion.