  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}.
