Is it feasible to break Diffie-Hellman key exchange when the implementation uses a poor-quality PRNG?
An attack would be trivial if the seed of the RNG was only 32 bits; just enumerate the seeds, and test which matches the intercepted messages. That's easy.

However the default Java Random class uses a 48-bit state and seed (which would still be attackable, though $2^{16}$ times less easily), and there are safe subclasses, thus use of Random does not imply an (insecure) 48-bit state, if the object really is a SecureRandom. See this Java 1.4.2 doc or this Java (1.)7.x one.

Update: While it seems the seed of the default Java Random class is 48-bit, its default seed indeed is 32-bit on some old Java platforms, and predictable to some degree on top of that; it is 48-bit on the latest Java platforms, and seems less predictable (though it could still be predictable/repeatable in a virtualized context).