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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 Cram\'er-Granville Conjecture to Honaker's
problem in a related context.

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