The Dragon Sword is a secret great sword that has very good stats, doubles in critical chance when your health is low and won't get lost on death, even if it's not equipped.

If you want to get the Dragon Sword, you need to unlock any skill that lets you extend your jumps. Since I mainly play with weapons of the "Blade" category (one-handed swords), the ideal skill for me was Air Puncture, which makes the character dash forward in an almost straight line.

Just before you face off against the Boss of the second level...
...turn right and approach the broken bridge with the shopping pillar...
... to the right, there is a small path that leads to a chest (not always a puzzle chest, rarely a purple Orb)...
   
... from that chest, if you look towards the other end of the broken bridge, you can already see the Dragon Sword stuck in the ground...

The distance from the chest to the other end is quite big, especially if you consider that parts of the visible remains of the bridge over there apparently have no mass (i.e. you can't stand on it and fall through, if you try).


In order to make it to the other side, I had to extend my jump as far as I could. As I mentioned above, I used a Blade (one-handed sword) for the jump and had the skill Air Puncture already unlocked.

With that setup, the easiest combination is as follows:

Jump forward, Light Attack, Strong Attack, Dodge, Strong Attack

With Mouse+Keyboard setup (default configuration):

Jump forward, [LMB], [RMB], [Shift], [RMB]

With an Xbox controller:

Jump forward, [X], [Y], [LT], [Y]

If you pull it off, you'll barely make it and be rewarded with the Dragon Sword.

Note that the Dragon Sword will always be in its original spot whenever you reload the level (by dying or quitting and reloading or after starting New Game+) but you won't get extra copies by picking it up again. If you have the Dragon Sword in your inventory and pick it up again, it will simply vanish.


Also be careful with the terrain over there, as most of it has no mass and you'll fall through if you venture too far, causing you to lose some health and to be stuck in the void below the world...
Don't worry, though. If that happens to you, just exit to the main menu and then continue. You'll be spawned right before the Boss area.

It is nevertheless possible to climb some of the first mountains on that side of the bridge, which can give you some nice vistas and allow you to reach previously unreachable places.