  This paper is motivated by the current development of several space missions
(e.g. ACES on International Space Station) that will fly on Earth orbit laser
cooled atomic clocks, providing a time-keeping accuracy of the order of
5~10^{-17} in fractional frequency. We show that to such accuracy, the theory
of frequency transfer between Earth and Space must be extended from the
currently known relativistic order 1/c^2 (which has been needed in previous
space experiments such as GP-A) to the next relativistic correction of order
1/c^3. We find that the frequency transfer includes the first and second-order
Doppler contributions, the Einstein gravitational red-shift and, at the order
1/c^3, a mixture of these effects. As for the time transfer, it contains the
standard Shapiro time delay, and we present an expression also including the
first and second-order Sagnac corrections. Higher-order relativistic
corrections, at least O(1/c^4), are numerically negligible for time and
frequency transfers in these experiments, being for instance of order 10^{-20}
in fractional frequency. Particular attention is paid to the problem of the
frequency transfer in the two-way experimental configuration. In this case we
find a simple theoretical expression which extends the previous formula (Vessot
et al. 1980) to the next order 1/c^3. In the Appendix we present the detailed
proofs of all the formulas which will be needed in such experiments.
