Measurements of $K^0$ and $\Lambda$ production in neutral current, deep inelastic scattering of 26.7 GeV electrons and 820 GeV protons in the kinematic range $10<Q^2<640$, $0.0003<x<0.01$, and $y>0.04$. Average multiplicities for $K^0$ and $\Lambda$ production are determined for transverse momenta $p_T >0.5> $ GeV and pseudorapidities $|\eta| < 1.3 $. The production properties of $K^0$ in events with and without a large rapidity gap with respect to the proton direction are compared. The ratio of neutral $K^0$ to charged particles per event in the measured kinematic range is, within the present statistics, the same in both samples.