Let E be a modular elliptic curve over a totally real number field F. We
prove the weak exceptional zero conjecture which links a (higher) derivative of
the p-adic L-function attached to E to certain p-adic periods attached
to the corresponding Hilbert modular form at the places above p where E has
split multiplicative reduction. Under some mild restrictions on p and the
conductor of E we deduce the exceptional zero conjecture in the strong form
(i.e.\ where the automorphic p-adic periods are replaced by the
\cL-invariants of E defined in terms of Tate periods) from a special case
proved earlier by Mok. Crucial for our method is a new construction of the
p-adic L-function of E in terms of local data
