One important potential application of high‐temperature superconductors is for tunnel junctions, especially hysteretic Josephson junctions. Unfortunately, tunnel junctions of oxide superconductors with a useful Tc≳20 K have exhibited nonideal I(V) characteristics with a normalized zero‐bias conductance, σs/σn(0), of 15–50%. We present data on point‐contact junctions of Ba1−xKxBiO3 which clearly demonstrate that these problems are not intrinsic. Using a normal‐metal counterelectrode, σs/σn(0) was <1% and σs/σn(V) agreed with the BCS theory by including only thermal smearing. Using a superconducting Nb counterelectrode, the I(V) showed very sharp features with evidence of a proximity effect, and σs/σn(0) was <0.2%. These results prove that there is no fundamental limitation for tunnel junctions using this HTS material, and one can anticipate operation of devices with two Ba1−xKxBiO3 electrodes at 15–20 K.