If the rate of the reaction is equal to the rate constant, the order of the reaction is -

Rate law: rate $= k[A]^n$, where n is the order.
If rate $= k$, then $[A]^n = 1$ for any concentration, which is possible only if $n = 0$.
For a zero order reaction, rate $= k[A]^0 = k$; the rate is independent of concentration.
So the order of the reaction is 0.