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We have measured, along the melting curve, the magnetic-field dependences of the velocities of the longitudinal and two transverse sound modes in the low field phase (U2D2) of nuclear-ordered solid 3He crystals with a single magnetic domain. We determined the elastic-stiffness tensors cij(T,B) along the melting curve at T=0.5 mK. Since the U2D2 phase has a tetragonal symmetry, we extracted six components of the nuclear-spin elastic-stiffnesses tensor Δc ijN(0.5 mK,B). We obtained the generalized Grüneisen constants Γij1/χ for the magnetic susceptibility χ from ΔcijN(0.5 mK,B), which were proportional to B2. [ABSTRACT FROM AUTHOR]