In contrast with the simultaneous structural and magnetic first order phase transition T0 previously reported, our detailed investigation on an underdoped Ba(0.84)K(0.16)Fe2As2 single crystal unambiguously revealed that the transitions are not concomitant. The tetragonal (τ: I4/mmm)-orthorhombic (ϑ: Fmmm) structural transition occurs at T(S)≃110 K, followed by an adjacent long-range antiferromagnetic (AFM) transition at T(N)≃102 K. Hysteresis and coexistence of the τ and ϑ phases over a finite temperature range observed by NMR experiments confirm the first order character of the τ-ϑ transition and provide evidence that both T(S) and T(N) are strongly correlated. Our data also show that superconductivity develops in the ϑ phase below T(c)=20 K and coexists with AFM. This new observation, T(S)≠T(N), firmly establishes another similarity between the hole-doped BaFe2As2 and the electron-doped iron-arsenide superconductors.