Summary: A Schwarzschild black hole being thermodynamically unstable, corrections to its entropy due to small thermal fluctuations cannot be computed. However, a thermodynamically stable Schwarzschild solution can be obtained within a cavity of any finite radius by immersing it in an isothermal bath. For these boundary conditions, classically there are either two black-hole solutions or no solution. In the former case, the larger mass solution has a positive specific heat and hence is locally thermodynamically stable. We find that the entropy of this black hole, including first-order fluctuation corrections, is given by: \[ \mathcal S=S_{\mathrm{BH}}-\ln\left[\frac 3R\left(S_{\mathrm{BH}}/4\pi\right)^{1/2}-2\right]^{-1}+\frac 12\ln(4\pi), \] where \(S_{\mathrm{BH}}=A/4\) is its Bekenstein-Hawking entropy and \(R\) is the radius of the cavity. We extend our results to four-dimensional Reissner-Nordström black holes, for which the corresponding expression is: \[ \mathcal S=S_{\mathrm{BH}}-\frac 12\ln\bigg[\left(\frac {S_{\mathrm{BH}}}{\pi R^2}\right) \left(\frac {3S_{\mathrm{BH}}}{\pi R^2}-\right. 2\sqrt{S_{\mathrm{BH}}/\pi R^2-\alpha^2}\left(\frac{\sqrt{S_{\mathrm{BH}}/\pi R^2}-\alpha^2} {(S_{\mathrm{BH}}/\pi R^2-\alpha^2)^2}\right))\bigg]^{-1} +\frac 12\ln (4\pi). \] Finally, we generalize the stability analysis to Reissner-Nordström black holes in arbitrary spacetime dimensions, and compute their leading order entropy corrections. In contrast to previously studied examples, we find that the entropy corrections in these cases have a different character.