Representing the notion of "until eternity" or "until further notice" is an iffy proposition.
Relational theory proper says that there is no such thing as null, so you're obliged to have whatever table it is split in two: one part with the rows for which the end date/end time is known, and another for the rows for which the end time is not yet known.
But (like having a null) splitting the tables in two will make a mess of your query writing too. Views can somewhat accommodate the read-only parts, but updates (or writing the INSTEAD OF on your view) will be tough no matter what, and likely to affect performance negatively no matter what at that).
Having the null represent "end time not yet known" will make updating a bit "easier", but the read queries get messy with all the CASE ... or COALESCE ... constructs you'll need.
Using the theoretically correct solution mentioned by dportas gets messy in all those cases where you want to "extract" a DATE from a DATETIME. If the DATETIME value at hand is "the end of (representable) time (billions of years from now as you say)", then this is not just a simple case of invoking the DATE extractor function on that DATETIME value, because you'd also want that DATE extractor to produce the "end of representable DATEs" for your case.
Plus, you probably do not want to show "absent end of time" as being a value 9999-12-31 in your user interface. So if you use the "real value" of the end of time in your database, you're facing a bit of work seeing to it that that value won't appear in your UI anywhere.
Sorry for not being able to say that there's a way to stay out of all messes. The only choice you really have is which mess to end up in.