The death star has a force field if we take everything literally, and those don't stop transporters. Plus, it only has it when it's near Endor's moon... Emitted by an unprotected installation.
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?
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.
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 and has a long recharge time. 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 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.
The borg wouldn't really defeat the death star, they'd assimilate it. It's too much of their kind of geometric ship not to. It would be fun to watch them with the stormtroopers. On one hand the stormtroopers aren't the brightest, in the other hand, their armor would probably be a little better against borg nanite probes. But would eventually all get assimilated.