We present first results for B-c spectroscopy using Lattice QCD. We us
e a non-relativistic effective action, matched to full QCD, which incl
udes spin-independent relativistic corrections. This approach represen
ts a considerable step beyond the potential models previously availabl
e. We use multi-exponential fits to multiple correlation functions to
extract ground and excited S states and give accurate values for the S
state hyperfine splitting and the P state (B-c*) fine structure, inc
luding the effects of P-1(1)/P-3(1) mixing.