In 1947 Mills proved that there exists a constant $A$ such that $\lfloor A^{3^n} \rfloor$ is a prime for every positive integer $n$. Determining $A$ requires determining an effective Hoheisel type result on the primes in short intervals--though most books ignore this difficulty. Under the Riemann Hypothesis, we show that there exists at least one prime between every pair of consecutive cubes and determine (given RH) that the least possible value of Mills' constant $A$ does begin with $1.3063778838$. We calculate this value to $6850$ decimal places by determining the associated primes to over $6000$ digits and probable primes (PRPs) to over $60000$ digits. We also apply the Cramér-Granville Conjecture to Honaker's problem in a related context.