By computer simulations, it was reported that the Bak-Tang-Wiesenfeld (BTW) model loses self-organized criticality (SOC) when some particles are annihilated in a toppling process in the bulk of system. We give a rigorous proof that the BTW model loses SOC as soon as the annihilation rate becomes positive. To prove this, a nonconservative Abelian sandpile model is defined on a square lattice, which has a parameter α (>~ 1) representing the degree of breaking of the conservation law. This model is reduced to be the BTW model when α= 1. By calculating the average number of topplings in an avalanche< T> exactly, it is shown that for any α> 1,< T><∞ even in the infinite-volume limit. The power-law divergence of< T> with an exponent 1 as α→ 1 gives a scaling relation 2 ν (2− a)= 1 for the critical exponents ν and a of the distribution function of T. The 1-1 height correlation C 11 (r) is also calculated analytically and we show that C 11 (r) is bounded by an exponential function when α> 1, although C 11 (r)∼ r− 2 d was proved by Majumdar and Dhar for the d-dimensional BTW model. A critical exponent ν 11 characterizing the divergence of the correlation length ξ as α→ 1 is defined as ξ∼| α− 1|− ν 11 and our result gives an upper bound ν 11<~ 1/2.