For each $N\ge c_dt^d$ we prove the existence of a spherical $t$-design on
the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending
only on $d$. This result proves the well-known conjecture of Korevaar and
Meyers concerning an optimal order of minimal number of points in a spherical
$t$-design on $S^d$ for a fixed $d$.