  We investigate the low-temperature phase diagram of the exactly solved su(4)
two-leg spin ladder as a function of the rung coupling $J_{\perp}$ and magnetic
field $H$ by means of the thermodynamic Bethe Ansatz (TBA). In the absence of a
magnetic field the model exhibits three quantum phases, while in the presence
of a strong magnetic field there is no singlet ground state for ferromagnetic
rung coupling. For antiferromagnetic rung coupling, there is a gapped phase in
the regime H < H_{c1}, a fully polarized gapped phase for H > H_{c2} and a
Luttinger liquid magnetic phase in the regime H_{c1} < H < H_{c2}. The critical
behaviour derived using the TBA is consistent with the existing experimental,
numerical and perturbative results for the strong coupling ladder compounds.
This includes the spin excitation gap and the critical fields H_{c1} and
H_{c2}, which are in excellent agreement with the experimental values for the
known strong coupling ladder compounds (5IAP)_2CuBr_4 2H_2 O, Cu_2(C_5 H_{12}
N_2)_2 Cl_4 and (C_5 H_{12} N)_2 CuBr_4. In addition we predict the spin gap
$\Delta \approx J_{\perp}-{1/2}J_{\parallel}$ for the weak coupling compounds
with $J_{\perp} \sim J_{\parallel}$, such as (VO)_2 P_2 O_7, and also show that
the gap opens for arbitrary $J_{\perp}/ J_{\parallel}$.
