We consider a one-dimensional spin model with the long-range random Hamiltonian given by H[sigma]=-1/2 & sum;(x not equal y)J(x,y)sigma(x)sigma(y)/|x-y|(alpha)0+(alpha)x, y. The randomness is considered in both the pairwise interaction J(x,y) and in its decaying parameter with slowest value alpha(0) plus a non-negative random variable alpha(x, y). We prove the loss of stability at alpha(0)=1/2. We also prove the existence of the free energy at the thermodynamic limit when alpha(0)>1/2. Furthermore, we show uniqueness of the equilibrium state for alpha(0)>3/2 in the strong sense.