The thermal conductivity of borocarbide superconductor LuNi2B2C was measured down to 70 mK (Tc/200) in a magnetic field perpendicular to the heat current from H = 0 to above Hc2 = 7 T. As soon as vortices enter the sample, the conduction at T → 0 grows rapidly, showing unambiguously that delocalized quasiparticles are present at the lowest energies. The field dependence is very similar to that of UPt3, a heavy-fermion superconductor with a line of nodes in the gap, and very different from the exponential dependence characteristic of s-wave superconductors. This is strong evidence for a highly anisotropic gap function in LuNi2B2C, possibly with nodes.PACS numbers: 74.70. Dd, 74.25.Fy, 74.60.Ec The vast majority of known superconductors are characterized by an order parameter with s-wave symmetry and a gap function which is largely isotropic and without nodes (zeros). Only four families of materials are seriously thought to exhibit a superconducting state with a different symmetry: (1) [4]. A major outstanding question is the nature of the microscopic mechanism responsible for superconductivity in any of these materials. The unconventional symmetry of the order parameter is evidence for a pairing caused by purely electronic interactions and not mediated by phonons. For example, the proximity to magnetic order which is found in all four families of superconductors has led to the suggestion that spin fluctuations are responsible for Cooper pairing, as is thought to be the case in superfluid 3 He. The presence of nodes in the gap function is generally associated with unconventional (non-s-wave) symmetries. These nodes are typically inferred from the observation of quasiparticle excitations at energies much lower than the gap maximum ∆ 0 , as reflected for example in the power law temperature dependence of various physical properties, such as London penetration depth and ultrasonic attenuation at T ≪ T c . Another way of detecting low-energy quasiparticles is to excite them by applying a magnetic field which introduces vortices in the material, so that the superfluid flow around each vortex Doppler shifts the quasiparticle energy. In certain limits, the quasiparticle response is the same whether induced by a thermal energy k B T or by a field energy ≃ ∆ 0 B/B c2 , where B c2 ≃ H c2 , the upper critical field [5].In this Letter, we turn our attention to another class of superconductors: the borocarbides LNi 2 B 2 C (where L = Y, Lu, Tm, Er, Ho, and Dy) [6]. It has generally been thought that these materials are described by an order parameter with s-wave symmetry and pairing which proceeds via the electron-phonon coupling [7][8][9]. However, there is recent evidence for low-energy excitations in the superconducting state, whether from the anomalous field dependence of the specific heat [10,11] and the microwave surface impedance [11,12], or from the presence of scattering below the gap in Raman measurements [13]. This has been interpreted in terms of an anisotropic s-wave gap [10,11] (see also reference [14]).Here w...