A medieval alchemist (from Europe) has figured out how to crystallize any object and turn it into diamond. Would diamond armor (e.g. chain mail and plate armor) and weapons be superior to their traditional counterparts? They would be lighter (density of diamond is 3.5gm/cm^3, roughly half of the density of steel/iron). However, I am concerned that since diamond armor would be so inflexible, it wouldn't absorb much of the energy of a hit.
The material property we'll want to concern ourselves here is Toughness. This value denotes a material's resistance to fracturing.
A typical diamond has a toughness value of 2.0 KIc, which is greater than most stones or rocks, and similar to ceramics. Steel, on the other hand, has a value of 50 KIc, and many metals have similarly high values compared to diamond.
In short, armor made from Diamond would likely shatter at first contact with a weapon, and would not be suited to use as a traditional armor of any kind.
That being said, there is some possibility that an armor that fractures upon receiving a blow might be useful against certain projectile weapons, since the impact would be distributed among the fractured pieces, but this protection would be a one-off, and there are ceramics more suited to such a purpose than diamond.
Diamond is a material with a very high hardness, but is not supple or flexible like metals.
By analogy, look at a traditional Katana. The blade is actually a complex three-dimensional structure, with levels of harness being created by controlling the tempering of the metal (for example, coating only part of the blade with clay when heating and quenching, so the coated parts have a different temper and are softer than the exposed cutting edge. (The hamon; wavy pattern along the side of a traditional Katana marks where the clay coating was applied.
A diamond edged blade made along the same principles would be incredibly sharp yet still be flexible enough to use without worrying too much about breakage. For movie fans, you could create something like the "Green Destiny" sword from "Crouching Tiger Hidden Dragon".
As for armour, pure diamond will suffer many of the same issues as using diamond as a blade. A diamond coating over steel plate or steel rings will make the armour stronger and more capable of resisting a strike by a weapon (diamond edged or not), but there could be issues of the diamond coating coming off the steel under layer due to the force of the blow. Possibly the best way to reduce this is to make your armour in scale form to allow for flexibility and absorb blows without separation
Diamond armor would have to be constructed very differently than traditional mail or plate armor for it to work in the manner intended.
As @MozerShmozer pointed out, Diamond has a very small toughness, and viability to cracking, so diamond plate armor is out. Like you stated in the OP, even a single layer of diamond mail would slowly disintegrate over multiple hits, even if scaled like @Thucydides suggested.
However, if you created armor with multiple layers of diamond scale, you would have something that is about the same weight as normal armor, with added protection from projectiles due to the diamond scales shattering away . The only issue that would arise is if a foe were to forcefully impact the exact same spot multiple times, which typically doesn't happen in a normal battle, until all the diamond scales in that one spot are gone. Depending on the thickness, a warrior with diamond scale armor could conceivably wear 3 or more layers (or even more layers on specific body locations) that would have the same weight of a typical armor of chain or plate mail.
Diamond may not be hugely useful as armor, but It would be fantastic for arrowheads, Javelin points, lance points or other "discard able" weapons.
the point (hah) of these weapons is that they only have to puncture a foe once, not to be used again and again. They should be good at holding an edge and puncturing things like leather and chain mail armor. This would be similar to flint and obsidian arrowheads that were fragile, yet incredibly sharp.
The possibility of shattering is actually attractive, as little tiny sharp bits are hard to find and extract bring the possibility of infection and sepsis.
Finally, can you imagine the shrapnel possibilities in a diamond medieval hand grenade? Yikes.
As other people have already mentioned, diamonds shatter too easily to be effectively used in most armor. You could maybe use it to some effect in scale armor, but I wouldn't trust it.
Like in the case of armor, diamonds are simply too easily fractured to be used in a medieval-style broadsword. However, if you are willing to bring in some non-medieval weaponry, it would be good in a Macuahuitl. A cross between a sword and a club, they are best described as "a cricket bat with a bunch of razor-sharp stones embedded in its sides". These stones are meant to break off and be replaced, while the main club remains intact. As such, it is immensely suitable to your purposes.