A CUR approximation of a matrix A is a particular type of low-rank approximation where C and R consist of columns and rows of A, respectively. One way to obtain such an approximation is to apply column subset selection to A and its transpose. In this work, we describe a numerically robust and much faster variant of the column subset selection algorithm proposed by Deshpande and Rademacher (2010), which guarantees an error close to the best approximation error in the Frobenius norm. For cross approximation, in which U is required to be the inverse of a submatrix of A described by the intersection of C and R, we obtain a new algorithm with an error bound that stays within a factor k+1 of the best rank-k approximation error in the Frobenius norm. To the best of our knowledge, this is the first deterministic polynomial-time algorithm for which this factor is bounded by a polynomial in k. Our derivation and analysis of the algorithm is based on derandomizing a recent existence result by Zamarashkin and Osinsky (2018). To illustrate the versatility of our new column subset selection algorithm, an extension to low multilinear rank approximations of tensors is provided as well.