  Let $\mathfrak g$ be a complex simple Lie algebra and let
$U_{\zeta}({\mathfrak g})$ be the corresponding Lusztig ${\mathbb
Z}[q,q^{-1}]$-form of the quantized enveloping algebra specialized to an
$\ell$th root of unity. Moreover, let $\mod(U_{\zeta}({\mathfrak g}))$ be the
braided monoidal category of finite-dimensional modules for
$U_{\zeta}({\mathfrak g})$. In this paper we classify the thick tensor ideals
of $\mod(U_{\zeta}({\mathfrak g}))$ and compute the prime spectrum of the
stable module category associated to $\text{mod}(U_{\zeta}({\mathfrak g}))$ as
defined by Balmer.
