Time would stop, and there would be nothing.
Let's first accept space as meaning everything physical. All universes, all matter, all physical properties. Let's also define the absence space as being the absence everything physical. Additionally, let's accept "nothing" as meaning the absence of all things physical, all properties, all ideas, all thoughts, and all concepts (because all of these things are "somethings") Let's also accept that our idea of measurement is not a tangible property and therefore, is irrelevant.
Now, we have a hard time thinking about time in the grand scheme, because we can't imagine what it would be like for time to be nonexistent. We can't imagine the absence of time because we think that the moment after it ceases is a new time in and of itself, but that's not necessarily true. The way I see it, when it stops, there will be no period of no time. We don't have to imagine what a period such as that would be like because the very idea of it is contradictory. So, if any such plausible, non-contradictory, circumstance could exist where time has stopped, it would have to be the end of the very last period of time to ever exist.
Now, if we are going theorize about a period where time still exists, but nothing else does (because like I said, a period of no time isn't a period at all), we have to accept that there will be no matter, no movement, no physical laws, and no measurement of space or distance. This is another thing we have trouble getting our minds around: the absence of size or measurement. This area of nonexistence isn't an area. It isn't a plane, or a void, or a realm, or a universe. The only thing that it is and that it possibly can be is a period of time. So, there is no visual image that we can conjure up that could represent this situation, because the presence of an image would make such a situation contradictory. Therefore, the absence of everything and the absence of existence with the exception of time is time itself.
So, we've concluded that there can two different circumstances in this sense: a circumstance where time doesn't exists and a circumstance where space doesn't exist. Now, consider a third. I believe this is the circumstance that you're asking about.
Imagine the end of a universe, or rather, the end of every universe. This can be happening over a period of an hour or over a period of trillions of years; it doesn't make a difference. Nevertheless, everything is ending at one rate or another. Eventually, each individual universe will disappear, leaving nothing, omitting time. Now, "nothing", as I've defined it, is the absence of everything. It's not the opposite of "everything", but rather, a word used to define the state in which "everything" is nonexistent. All you have left at this point is time, ticking onwards. Now, imagine that time stops. A circumstance of no time, as I have described, is simply the end of all time. All you have left at this point is nothing.
But, there's a problem and that is, something still exists ("nothing"). If we define "nothing" as being the absence of all properties, then we have to take "nothing" into question on whether it, in and of itself, is a property. In the manner that I've defined it, it is. It is the state of the absence of everything. It is the property that dictates that nothing can exist within a situation of nothingness. (And properties can't exist in a circumstance of nonexistence because they are tangible and therefore, if properties exist, then your circumstance merely becomes a circumstance of no time and no matter. This is not nonexistence.)
Once time and space have stopped, there is no plausible way to remove "nothing." Ending time removes time and ending space removes the physical, but "nothing" is the un-removable property. This is because "nothing" does not just apply to the physical, but also, the theoretical. It relates to time, as well, and the absence time in necessary for the presence of nothing. "Nothing" has been preserved after the end of all space by the passing of time. Since time's destruction does not include the destruction of properties, it can not destroy "nothing".
Therefore, all existence cannot cease. By this logic, nonexistence is possible only if not only space, but time has never existed, but it impossible to reach after a period of existence.