We present high-resolution radio-continuum observations of the nascent starburst in the metal-poor galaxy SBS 0335−052. These radio data were taken with the Very Large Array and include observations at 0.7 cm, 1.3 cm, 2 cm, 3.6 cm, and 6 cm. These observations enable us to probe the thermal radio nebulae associated with the extremely young star-forming regions in this galaxy. Two discrete and luminous star-forming regions are detected in the south of the galaxy that appear to be associated with massive star clusters previously identified at optical wavelengths. However, the remaining optically identified massive star clusters are not clearly associated with radio emission (either thermal or nonthermal) down to the sensitivity limits of these radio data. The spectral energy distributions (SEDs) of the two radio-detected clusters are consistent with being purely thermal, and the entire region has an inferred ionizing flux of ∼1.2 × 10 53 s −1 , which is equivalent to ∼12,000 "typical" O-type stars (type O7.5 V). The observations presented here have resolved out a significant contribution from diffuse nonthermal emission detected previously, implying a previous episode of significant star formation. The current star formation rate (SFR) for this southern region alone is ∼1.3 M yr −1 , or ∼23 M yr −1 kpc −2 , which is nearing the maximum starburst intensity limit. This SFR derived from thermal radio emission also suggests that previous optical recombination line studies do not detect a significant fraction of the current star formation in SBS 0335−052. From model fits to the radio SED, we infer a global mean density in the two youngest clusters of n e 10 3 -10 4 cm −3 . In addition, a comparison between the compact and diffuse radio emission indicates that up to ∼50% of the ionizing flux could be leaking out of the compact H ii regions; this in is agreement with previous work, which suggests that the interstellar medium surrounding the natal clusters in SBS 0335−052 is porous and clumpy.