In this paper, we analyze a special immersed finite volume method that is different from the classic immersed finite volume method by choosing special control volumes near the interface. Using the elementwise stiffness matrix analysis technique and the $H^1$-norm-equivalence between the immersed finite element space and the standard finite element space, we prove that the special finite volume method is uniformly stable independent of the location of the interface. Based on the stability, we show that our scheme converges with the optimal order $\mathcal{O}$($h$) in the $H^1$ space and the order $\mathcal{O}$($h^{3/2}$) in the $L^2$ space. Numerically, we observe that our method converges with the optimal convergence rate $\mathcal{O}$($h$) under the $H^1$ norm and with the the optimal convergence rate $\mathcal{O}$($h^2$) under the $L^2$ norm all the way even with very small mesh size $h$, while the classic immersed finite element method is not able to maintain the optimal convergence rates (with diminished rate up to $\mathcal{O}$($h^{0.82}$) for the $H^1$ norm  error  and  diminished  rate  up to $\mathcal{O}$($h^{1.1}$) for $L^2$-norm error), when $h$ is getting small, as illustrated in Tables 4 and 5 of [35].