The pilot does sit in a sphere. It's just like the Trieste; it's not possible to make a sphere that is strong enough to withstand the pressure and light enough to remain buoyant. To solve this they attach the sphere to a larger buoyant vessel.
The non-spherical part of the Trieste was filled with gasoline for buoyancy, this submersible uses some kind of high density foam.
The foam can be high density and still be less dense than water. They would have to use foam that wasn't easily compressed or the structure would collapse from the pressure.
edit--I just checked, the type of foam used ranges from about 20 to 45 lbs/ft³. That qualifies as high density.
Since I was curious about the relative densities, I looked it up and asked Google to convert to your archaic units. Water, at -30C, is around 60 lbs/ft³, so indeed the foam is denser.
I was _going_ to claim that water is denser still, way down deep in the ocean. But I thought I'd check it out first, and although I'm not well-educated enough to understand the important parts of phrases like "At 0 °C, at the limit of zero pressure, the compressibility is 5.1×10−10 Pa−1", I can understand this phrase: "The low compressibility of water means that even in the deep oceans at 4 km depth, where pressures are 40 MPa, there is only a 1.8% decrease in volume."
Oh, my bad. I misrecalled a detail from the featured article, thinking it had claimed that temperatures were very very low, so I picked the lowest temperature in that sidebar as a rough approximation. Actually, TFA says "near freezing", which certainly much closer to 4C than to -30C.
The non-spherical part of the Trieste was filled with gasoline for buoyancy, this submersible uses some kind of high density foam.