  Let $\lfloor x\rfloor$ denote the integer part of $x$. In 1947, Mills
constructed a real number $\xi$ greater than $1$ such that $\lfloor \xi^{3^k}
\rfloor$ is always a prime number for every positive integer $k$. We define
Mills' constant as the smallest real number $\xi$ satisfying this property. In
this article, we determine that Mills' constant is irrational. Moreover, we
also obtain partial results on the transcendency of the constant.
