The magnetic, magnetocaloric and thermal characteristics have been studied in a Ni(50.3)Mn(20.8)Ga(27.6)V(1.3) ferromagnetic shape memory alloy (FSMA) transforming martensitically at around 40 K. The alloy shows first a transformation from austenite to an intermediate phase and then a partial transformation to an orthorhombic martensite, all the phases being ferromagnetically ordered. The thermomagnetization dependences enabled observation of the magnetocaloric effect in the vicinity of the martensitic transformation (MT). The Debye temperature and the density of states at the Fermi level are equal to θ(D) = (276 ± 4) K and 1.3 states/atom eV , respectively, and scarcely dependent on the magnetic field. The MT exhibited by Ni-Mn-Ga FSMAs at very low temperatures is distinctive in the sense that it is accompanied by a hardly detectable entropy change as a sign of a small driving force. The enhanced stability of the cubic phase and the low driving force of the MT stem from the reduced density of states near the Fermi level.