Let $k$ be a field of characteristic $0$ (not necessarily algebraically closed), and let $A=k[x^1,\ldots,x^m]/(f_1,\ldots,f_N)$ be an affine variety which is a complete intersection; i.e. $\dim(A)=m-N$. (For definiteness, let's assume that $m-N\geq 3$.)
Is it always possible to find an affine hypersurface $B=k[x^1,\ldots,x^m]/(c)$ such that the intersection with $A$ has dimention $m-N-1$ without adding any new singularities (i.e. compared to the singularities of $A$)?