I have been browsing the internet, and I have started to wonder whether a diamond sword is feasible or not. I would very much like to know the answer.
You could, but it probably wouldn't be worth the effort. Yes, diamond is hard, but unless you have an ABSOLUTELY PERFECT crystalline structure all the way down to the nanometer scale, any flaw, even a tiny scratch, would eventually turn into a massive crack as soon as you started applying impact force. You could put a fantastic edge on a diamond sword, but as soon as you started hitting things with it, it'd crack and shatter pretty quickly.
Steel makes good swords because it's hard enough to take an edge, but soft enough to absorb impact without shattering. A major part of swordmaking is striking the proper balance between the two.
How about a cheaper route?
The Macuahuitl, the "Aztec Sword" was pretty much a Cricket bat, but with obsidian blades inserted into the edges. If a blade cracked or was broken beyond resuse, you could just replace it.
Following that idea, you could have a wooden (or metallic) handle and middle-part of the blade, and then have a diamond edge. While this is not a pure diamond sword, you do not need to worry about micro-fractures nearly as much, and repairing this weapon is a viable option, compared to a single diamond being the sword.
In practical terms, the other answers cover the macro scale - either a brittle blade of pure diamond (maybe used only for ceremonial purposes or special executions). Or a diamond coated blade.
On a much smaller scale, diamond blades are used in laboratories, not as grit for grinding purposes - but as single atom thick (ideally, but in practice a few atoms thick at the edge) blades for slicing samples real thin - or as a diamond knife in a surgical procedure called radial keratomy (since the 1950's).
The bad-news, 6 mm wide blades (0.236 inches) start at over $2,000 each (beware - commercial link). As swords go, this is even smaller than "handy pocket-sized".
Diamonds, mostly industrial quality and not gem quality, are used for cutting. As I remember, there are diamond edged saws for various industrial uses, and the diamonds on the edges have to be replaced as they wear out.
So I can picture someone using a technological or magical sword with a chainsaw action and many tiny diamonds on the edges to improve cutting through armor or flesh or whatever.
Maybe it cuts through anything and everything at the start of the battle but as it cuts through more and more armor and bodies the diamonds crack and shatter and the edge is reduced to mere steel and it only cuts as well as an ordinary chainsaw would.
Short Answer. No
Most of the detail on why this will not work is covered in this question.
While it may be possible to create a sword shaped object from diamond it is simply too brittle to be used effectively.
On a side note a sword of pure carbon would also lack the mass to be effective against a similar sized steel blade. In swordplay inertia is often your friend.
The subject of alternative materials has also been covered in this question