We study lattice vertex algebras of type ADE over an algebraically closed field of prime characteristic $p>2$. When $p$ does not divide the Gram determinant, we establish an isomorphism with the level-one simple affine vertex algebra of the same type and classify the irreducible $\mathbb N$-graded modules. For type $A_n$ with $p\mid n+1$, we determine the radical of the standard bilinear form and identify the simple quotient with the level-one simple affine vertex algebra associated with $\mathfrak{psl}_{n+1}$. We also classify the irreducible $\mathbb N$-graded modules of the original algebra and its simple quotient. This talk is based on joint work with Hongju Zhao.