We report the detection of the double neutron star binary (DNSB) B1534+12 (= J1537+1155) with the Chandra X-Ray Observatory. This DNSB (P-orb = 10.1 hr) consists of the millisecond (recycled) pulsar J1537+1155A (P-A = 37.9 ms) and a neutron star not detected in the radio. After the remarkable double pulsar binary J0737-3039, it is the only other DNSB detected in X-rays. We measured the flux of (2.2 +/- 0.6) x 10(-15) ergs s(-1) cm(-2) in the 0.3-6 keV band. The small number of collected counts allows only crude estimates of spectral parameters. The power-law fit yields the photon index Gamma = 3.2 +/- 0.5 and the unabsorbed 0.2-10 keV luminosity L-X approximate to 6 x 10(29) ergs s(-1) approximate to 3 x 10(-4) E-A, where E-A is the spin-down power of J1537+1155A. Alternatively, the spectrum can be fitted by a blackbody model with T approximate to 2.2 MK and the projected emitting area of similar to 5 x 10(3) m(2). The distribution of photon arrival times over binary orbital phase shows a deficit of X-ray emission around apastron, which suggests that the emission is caused by interaction of the relativistic wind from J1537+1155A with its neutron star companion. We also reanalyzed the Chandra and XMM-Newton observations of J0737-3039 and found that its X-ray spectrum is similar to the spectrum of B1534+12, and its X-ray luminosity is about the same fraction of E-A, which suggests similar X-ray emission mechanisms. However, the X-ray emission from J0737-3039 does not show orbital phase dependence. This difference can be explained by the smaller eccentricity of J0737-3039 or a smaller misalignment between the equatorial plane of the millisecond pulsar and the orbital plane of the binary.