If \(A\) is an associative PI-algebra with 1 in characteristic zero, then the codimension sequence \(c_n(A)\) is asymptotic to a function of the form \(\alpha n^t C^n\), where \(C\) is a positive integer [the first author, Adv. Appl. Math., 41, 52--75 (2008; Zbl 1145.05052)]. Following Kemer, the most important PI-algebras are the verbally prime ones, which serve as basic building blocks in his theory. There are three families of verbally prime PI-algebras: \(M_ k (F)\), matrices over the field; \(M_{k,l}\), the Kemer algebras; and \(M_k (E)\), matrices over the infinite dimensional Grassmann algebra \(E\). The asymptotics of the codimensions have been studied in all three cases. In the case of \(M_k(F)\) all three constants: \(\alpha, t, C\) are known. In the case of \(M_{k,l}\) the constant \(\alpha\) is not known, but the other two parameters are known. In case of \(M_k (E)\) the number \(C\), the exponent, was known. The goal of the paper is to give the degree of the polynomial factor. Namely, the authors establish the asymptotic \[c_n(M_k(E))\simeq \alpha n^{(1-k^2)/2}(2k^2)^n, \] where the constant \(\alpha\) remains unknown.