We predict the 21-cm signature of the first metal-free stars. The soft X-rays emitted by these stars penetrate the atomic medium around their host halos, generating Lyman alpha photons that couple the spin and kinetic temperatures. These creates a region we call the Lyman alpha sphere, visible in 21-cm against the CMB, which is much larger than the HII region produced by the same star. The spin and kinetic temperatures are strongly coupled before the X-rays can substantially heat the medium, implying that a strong 21-cm absorption signal from the adiabatically cooled gas in Hubble expansion around the star is expected when the medium has not been heated previously. A central region of emission from the gas heated by the soft X-rays is also present although with a weaker signal than the absorption. The Lyman alpha sphere is a universal signature that should be observed around any first star illuminating its vicinity for the first time. The 21-cm radial profile of the Lyman alpha sphere can be calculated as a function of the luminosity, spectrum and age of the star. For a star of a few hundred solar masses and zero metallicity (as expected for the first stars), the physical radius of the Lyman alpha sphere can reach tens of kiloparsecs. The first metal-free stars should be strongly clustered because of high cosmic biasing; this implies that the regions producing a 21-cm absorption signal may contain more than one star and will generally be irregular and not spherical, because of the complex distribution of the gas. We discuss the feasiblity of detecting these Lyman alpha spheres, which would be present at redshifts $z\sim 30$ in the Cold Dark Matter model. Their observation would represent a direct proof of the detection of a first star.

replaced with ApJ accepted version. Many minor revisions and additional references, major results unchanged