One relatively tedious (but straightforward) solution would be to break the big database up into smaller databases, do the conversion on the smaller databases, and then recombine them.
This has an added benefit that if, by some chance, the text is an invalid date in one chunk, it will be easier to find (because of the smaller chunk sizes).
Assuming you have some kind of integer key on the table that ranges from 1 to (say) 10000000, you can just do queries like
SELECT *
INTO newTable1
FROM yourtable
WHERE yourkey >= 0 AND yourkey < 1000000
SELECT *
INTO newTable2
FROM yourtable
WHERE yourkey >= 1000000 AND yourkey < 2000000
etc.
Make sure to enter and run these queries seperately, since it seems that Access will give you a syntax error if you try to run more than one at a time.
If your keys are something else, you can do the same kind of thing, but you'll have to be a bit more tricky about your WHERE clauses.
Of course, a final thing to consider, if you can swing it, is to migrate to a different database that has a little more power. I'm guessing you have reasons that this isn't easy, but with the amount of data you're talking about, you'll probably be running into other problems as well as you continue to use Access.
EDIT
Since you are still having some troubles, here is some more detail in the hopes that you'll see something that I didn't describe well enough before:
Here, you can see that I've created a table "OutputIDrive" similar to what you're describing. I have an ID tag, though I only have three entries.
Here, I've created a query, gone into SQL mode, and entered the appropriate SQL statement. In my case, because my query only grabs value >= 0 and < 2, we'll just get one row...the one with ID = 1.
When I click the run button, I get a popup that tells/warns me what's going to happen...it's going to put a row into a new table. That's good...that's what we're looking for. I click "OK".
Now our new table has been created, and when I click on it, we can see that our one line of data with ID = 1 has been copied over to this new table.
Now you should be able to just modify the table name and the number values in your SQL query, and run it again.
Hopefully this will help you with whatever tripped you up.
EDIT 2:
Aha! This is the trick. You have to enter and run the SQL statements one at a time in Access. If you try to put multiple statements in and run them, you'll get that error. So run the first one, then erase it and run the second one, etc. and you should be fine. I think that will do it! I've edited the above to make it clearer.