The scenario as phrased suggests the Borg as a whole, not one cube. To which the answer is The Borg, obviously.
I think the main weakness the Death Star has in the scenario where it goes up against just one cube is that its weapon takes a significant amount of time to fire.
If it's the Borg as a whole, then shouldn't it also be the entire Imperial Navy? Not just the Death Star, but Super Star Destroyers and the entire rest of their fleet?
Even then, the Imperial Navy doesn't exactly have a large variety in weaponry. Blasters, bombs, and maybe one guy with a lightsaber if he feels like it. The Turbolaser of the Death Star was just a scaled up version of a blaster. The Borg would transport a small team up and adapt within seconds. It's a non-fight after that against the whole Navy, even for just one Cube.
The Borg don't have empathy or a worry about planets being destroyed, so the death star is not a compelling weapon against them.
Plus, the death star can only fire one beam and then had to recharge. During that time the Borg could assault the surface of the death star and their shield definitely would adapt to the short range canons.
It all comes down to whether the death star uses a force field to defend itself and how much power it can generate vs the borgs assault.
I don’t know, I remember when they chased the Enterprise they just kept hitting it with shots designed to dampen their shields, but they didn’t want to destroy the ship and kill Picard.
How would the Death Star hit the cube in the first place though? It can travel faster than light, whereas the Death Star is meant for stationary or nearly stationary targets. Everything else – the fighters and the turrets – the cube would adapt to immediately. The Death Star 2 is shown to have a shield around it projected from Endor, but the first Death Star is never shown to have such a shield. So unless the cube gets in firing range of the main cannon, it just adapts immediately to everything and starts neutering the station's defenses before beginning the assimilation.
I'm not sure that last part's true. The Death Star's beam is pretty wide, and it manages to track a planet, which if we take Earth's travel around the Sun as an example could be moving around 30 km/s. Voyager 1, by contrast, travels around 17 km/s. The main thing it would depend on is your distance from the Death Star. The laser seems to be able to travel a very long distance, and so if you're far enough away, that only requires a very small angular correction by the station, although they'd have to lead the shot more and would give the ship more chances to make unpredictable maneuvers after they've already fired. Collimation means that the beam would stay at basically the same width when it reaches you as it was at the station, so I don't think that would be a factor. If you're really close, though, then yeah, even just lazily moving out of the way is probably enough. I don't think it's ever quite explained how the Death Star tracks Alderaan, but based on the fact that the laser is facing like 150 degrees away from the planet as Princess Leia is brought in for interrogation, I think it can be surmised that it can orbit itself about its own axis pretty quickly.
I agree with every argument except one. I don't see how the Borg's ships speed changes anything. The starships and starfighters in Star Wars also travel at warp speeds.
I’m not sure what the original commenter was thinking, but they have assimilated the Picard Manoeuvre and they could trivially kite the death star with it.