We performed the first scanning tunneling spectroscopy measurements on the pyrochlore superconductor KOs2O6 (Tc = 9.6 K) in both zero magnetic field and the vortex state at several temperatures above 1.95 K. This material presents atomically flat surfaces, yielding spatially homogeneous spectra which reveal fully-gapped superconductivity with a gap anisotropy of 30%. Measurements performed at fields of 2 and 6 T display a hexagonal Abrikosov flux line lattice. From the shape of the vortex cores, we extract a coherence length of 31-40Å, in agreement with the value derived from the upper critical field Hc2. We observe a reduction in size of the vortex cores (and hence the coherence length) with increasing field which is consistent with the unexpectedly high and unsaturated upper critical field reported.PACS numbers: 74.70. Dd, 74.50.+r, 74.25.Qt The discovery of superconductivity in the β-pyrochlore osmate compounds AOs 2 O 6 (A = K, Rb, Cs) [1] has highlighted the question of the origin of superconductivity in classes of materials which possess geometrical frustration [2,3]. Interest has been predominantly focused on the highest-T c compound KOs 2 O 6 which presents many striking characteristics. In particular, the absence of inversion symmetry in its crystal structure [4] raises the question of its Cooper pair symmetry and the possibility of spin singlet-triplet mixing [5,6].The pyrochlore osmate compound KOs 2 O 6 displays a critical temperature T c = 9.6 K, the largest in its class of materials (CsOs 2 O 6 and RbOs 2 O 6 which differ only by the nature of the alkali ion have T c s of 3.3 and 6.3 K respectively). Although band structure calculations show that the K ion does not influence the density of states (DOS) at the Fermi level [7,8], it seems to affect several key properties [9]. In particular, the first order phase transition revealed by specific heat measurements in magnetic fields at the temperature T p ≈ 7.5 K has been ascribed to a "freezing" of its rattling motion [10]. The negative curvature of the resistivity as a function of temperature also indicates a large electron-phonon scattering [11]. Specific heat measurements [12] suggest the coexistence of strong electron correlations and strong electron-phonon coupling, two generally antagonistic phenomena with respect to the superconducting pairing symmetry. The nature of the symmetry remains a controversial subject in the literature. NMR [13] and µSR [14] data suggest anisotropic gap functions with nodes whereas thermal conductivity experiments [15] favor a fully-gapped state.The peculiar behavior of KOs 2 O 6 is demonstrated by its upper critical magnetic field H c2 , whose temperature dependence is linear down to sub-Kelvin temperatures and whose amplitude is above the Clogston limit [16]. One possible interpretation is the occurrence of spin-triplet superconductivity driven by spinorbit coupling [5,6]. Alternatively, it has also been suggested that this behavior can be explained by the peculiar topology of the Fermi surface (FS) sheets of KOs ...