We present 21-cm absorption measurements towards 12 radio continuum sources with previously identified thermally-unstable warm neutral medium (WNM). These observations were obtained with the Expanded Very Large Array (EVLA) and were complemented with the HI emission spectra obtained with the Arecibo Observatory. Out of 12 sources, HI absorption was detected along 5 lines of sight (seven new absorption features in total), resulting in a detection rate of ~42%. While our observations are sensitive to the WNM with a spin temperature T_s<3000 K, we detected only two wide absorption lines with T_s=400-900 K. These temperatures lie above the range allowed for the cold neutral medium (CNM) by the thermal equilbrium models and signify the thermally unstable WNM. Several absorption features have an optical depth of only a few x10^{-3}. While this is close or lower than what is theoretically expected for the CNM, we show that these weak lines are important for constraining the fraction of the thermally unstable WNM. Our observations demonstrate that, for the first time, high bandpass stability can be achieved with the VLA, allowing detection of absorption lines with a peak optical depth of ~10^{-3}.
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