The author gives a refinement of \textit{V. A. Kondrat'ev}'s result on boundary value problems for elliptic equations in a domain with conic corners [Tr. Mosk. Mat. O.-va 16, 209--292 (1967; Zbl 0162.16301)]. Let \(K\) be the angle on the plane \(x=(x_1,x_2):00\); \(\gamma_ 1:\theta=\theta_ 0\), \(r>0\), \(r=\sqrt{x_1^2+x_2^2}\). Denote by \(V^\ell_{2,\beta}(K)\) the space of functions obtained from \(C_0^ \infty(\bar K\backslash 0)\) by completion with respect to the norm \[ \| u\|_{V^ \ell_{2,\beta}}=\left (\sum_{|\alpha|\leq\ell}\int_ Kr^{2(\beta- \ell+|\alpha|)}| D^ \alpha u|^2\,dx\right)^{1/2}. \] Consider the following Neumann problem in the angle \(K\): \[ -\Delta u=f, \quad x\in K, \quad (\partial u/\partial n)|_{\gamma_0}=\varphi_0, \quad (\partial u/\partial n)|_{\gamma_1}=\varphi_1, \leqno (1) \] where \(n\) is the exterior normal to \(\partial K\). Then the main assertion of the author is the following: Let \(\ell=0,1,\ldots\) and the real number \(\beta\) differs from \(1-\pi k/\theta_ 0\), \(k=0,\pm1,\ldots,\) then for any \(f\in V^ \ell_{2,\beta+\ell}(K)\), \(\varphi_ j\in V^{\ell+1/2}_{2,\beta+\ell}(\gamma_ j)\), \(j=0,1,\) the problem (1) has a unique solution, belonging to the space \(V^{\ell+2}_{2,\beta+\ell}(K)\) and the inequality \[ \| u\|_{V^{\ell+2}_{2,\beta+\ell}}\leq c(\| f\|_{V^ \ell_{2,\beta+\ell}(K)}+\sum^1_{j=0}\|\varphi_ j\|_{V^{\ell+1/2} _{2,\beta+\ell}(\gamma_ j)}) \] holds with constant \(c\) not depending on \(\varphi_ j\).