Question is in the title. No Rule 34 jokes, please.
As the proud part-owner (well, part-creator anyway) of the object under discussion, I can tell you that this is typically determined by an ultra-sound scan. In the UK, these are usually done around weeks 12 and 20 of the pregnancy. At the 12-week scan, you can just about make out the head and possibly some limbs - determining gender is out of the question. That is usually offered at the 20-week scan, though even then the baby has to be in the right position and they often can’t be sure. So presumably the answer is somewhere around the 16-week mark as an absolute minimum. No doubt someone with actual medical knowledge will be along shortly.
As for us, my wife didn’t want to find out, so we didn’t - they always ask at the beginning of the 20-week scan. At the end my wife asked the sonographer if they saw anything - they said no, if they are told not to check, they just don’t look. I guess you have to manipulate the scanning device in a certain way to get the correct angle, though of course sometimes the answer becomes obvious by accident - didn’t with us though.
Presumably it would be possible to determine the gender very early on indeed with some sort of invasive test, but obviously that isn’t done because it risks harming the fetus unnecessarily. Hope this helps.
Maternal blood tests can yield fairly reliable results as early as 7 weeks. Amniocentesis, which gives a pretty complete picture of fetal DNA, can be done at around 15 weeks, but it carries some risk of miscarriage, so it’s not generally recommended unless there’s some increased risk of genetic defects.
At 12 weeks, we had a blood test (mother’s blood) that collected fetal DNA as part of a genetic screening, and a “side effect” of that is the doctor was able to tell us the sex of the child.
If you do PGD as part of IVF, you know the sex before the embryos are put back in to the uterus (so around 3 to 5 days past conception).
This was my first thought when I read the title line. BTW, when it is put back in the uterus is it an embryo or a blastocyst or a zygote?
Depends on how far along it is. If it’s a 2- or 3-day transfer, it’s called an embryo. If it’s day 5 or 6, it’s usually a blast. It can be called an expanding/hatching blast too, depending on if it’s, well, hatching.
5:45 am, as long as they’ve had their coffee.
A colleague of mine was told, based on multiple ultrasounds, that he would be getting a daughter. Then, in the last ultrasound which was done just a few weeks before the delivery, it turned out that the child was male after all. Apparently, ultrasounds can be deceiving even for the experts.
My former stepmother was in a fender-bender while pregnant with my future half-brother. She considered (as a joke) suing the other driver for making her unborn child turn from a girl into a boy. It was all in jest, of course – she’s a lawyer, and apparently her claim was no less crazy than the cases she had to deal with every day. 
A question about maternal blood tests & genetic screening – are these done as a matter of routine, or do you need to request them directly? My sister-in-law is three months along, but she doesn’t want to know the child’s gender yet; I’m just wondering if her doctor may already know.
Ours was done as a matter of routine, possibly age related (just over 35), but we were given an option IIRC of whether we wanted to do the screen and definitely when the doctor gave us the results, he asked us directly if we wanted to know the sex. I can’t remember the name of the company/test, but the sex determination is supposed to be 99.4% accurate.