For the directed polymer in a random environment (DPRE), two critical inverse-temperatures can be defined. The first one, $β_c$, separates the strong disorder regime (in which the normalized partition function $W^β_n$ tends to zero) from the weak disorder regime (in which $W^β_n$ converges to a nontrivial limit). The other, $\bar β_c$, delimits the very strong disorder regime (in which $W^β_n$ converges to zero exponentially fast). It was proved previously that $β_c=\bar β_c$ when the random environment is upper-bounded for the DPRE based on the simple random walk. We extend this result to general environment and arbitrary reference walk. We also prove that $β_c=0$ if and only the $L^2$-critical point is trivial.