They had to go to the Stargate people to figure that out? I would have answered that immediately.

The first Borg cube was easily attacked and damaged by a much less powerful starship and the Borg themselves were initially overcome by Federation weapons.

But not for long.

Of course the Death Star would win against a Borg cube and of course it would only happen once.

Also, I don't know if there are anything near 1:1 power matches in Star Wars vs. Star Trek.

So, here's a thought; the Death Star's superlaser is the only real laser weapon in Star Wars, while basically every other ranged weapon (blaster) is a plasma weapon. The main strength of the borg's shields is that they're able to adapt to phaser frequency, and while phasers are a completely fictional technology, they're definitely not plasma. Do we know how well a borg cube's shields would hold up against plasma? Is it possible that they'd go down to a barrage of Tie Fighter and Blaster Cannon fire, without the need to even use the superlaser?

Edit: OK, so I went down a rabbit hole on this, and this is what I learned about the Borg's Adaptive Shield Matrix:

All phasers are generated on a particular subspace phase compression pulse frequency, whilst torpedo warheads all possess their own shielding which also possesses its own subspace phase compression pulse frequency. Adaptive Shielding works by remodulating the shields to the identical subspace compression pulse frequency of torpedos and phasers...

So, it seems like the Borg's sheilds adapt to the subspace pulse frequencies of phasers and torpedos. Phasers are a type particle weapon that Gene Roddenberry made up when he realized lasers didn't work the way he thought they did, and they don't really have much basis in the real world like lasers or plasma weapons do.

Since the Adaptive Shield Matrix specifically works by adapting to subspace frequencies, there's really no reason to think that their shields would have a distinct advantage over lasers or plasma bolts the way they do phasers. The fact the Picard was able to easily kill some Borg with hard-light bullets seems to back this up.

So, if the Borg shields don't nullify the Death Star's weapons like they do Star Trek starships, this just comes down firepower. The superlaser should be able to destroy any Borg cube multiple times, and even without the superlaser, they're massively outgunned. This is 3000 meter ship against a 75 mile wide battle station. Even if the 10,000 turbolaser, 2,500 laser cannons, 2,500 ion cannons can't overpower them (and by the way, it sounds like at least some of those, "lasers," are actually plasma weapons based on wookiepedia, because of course Star Wars can't be consistent), and the 768 tractor beam projectors can't immobilize the cube, the 7,000 individual tie fighters would probably overwhelm the it. Hell, if the Death Star is faster, they could probably just smash into them and still survive the damage.

I think the Death Star has this by a mile. I hate to admit it, but I don't see a win condition for the Borg here.

I think, while the death star could probably take out a borg cube, as mentioned by others with the size discrepancy alone, I think it becomes a potentially harder calculus if you field even two, potentially more, borg cubes. At that point, it sort of falls down to how the borg shields hold up against blasters and ion cannons, and that seems like kind of a shitfuck. There's not a real answer to it, because both of them are either totally fictional or theoretical technologies. That maybe also holds true for things like the death star's laser, but as others have said, there's better analogues for that which we see the borg go up against and lose.

I think there's also potentially an interesting difference here in how the Borg's warp drive works vs Star wars hyperdrives. Could the Borg cube just jump right up to the death star and then board immediately? Could they bypass the laser by just warping in? I dunno, also a consideration, also unanswerable.

Would the death star be hackable? I dunno, the star wars computers might be too different, or even just too fucking old. They seem more fucked up than the star trek computers, anyways.

I dunno. They probably both get wiped by the halo ring, in any case.

Why is everyone assuming that the death star would use it's superlaser (apparently that's what it's called) against the borg cube? The superlaser has a very low fire rate and I doubt it's good in combat against any small moving target, and to the death star that's what a borg cube is. The general armament of a death star seems completely useless against a borg cube because it would adapt to the regular turbo lasers almost immediately.

Whoever said this doesn't know either setting very well. Borg become resistant, not immune to weapons they encounter. Starships were still perfectly capable of damaging Borg ships with the same weapons years later.

The problem is that the two settings operate on entirely different scales. Individual Turbolaser bolts are something like thousands of 24th century Photon Torpedoes? Even if we assume the Cube can resist 90% of the energy from each bolt, it's still having a very bad day.

What would the Borg do if Vader boarded their cube and went all Vader on them?

I lean more towards being a fan of Star Trek than Star Wars, but Vader is pretty powerful.