Distance to normal elements in $C^{\ast }$-algebras of real rank zero

We obtain an order sharp estimate for the distance from a given bounded operator $A$ on a Hilbert space to the set of normal operators in terms of $\mathrm{\parallel }[A,A^{\ast }]\mathrm{\parallel }$ and the distance to the set of invertible operators. A slightly modified estimate holds in a general $C^{\ast }$-algebra of real rank zero.