to execute a will = ausfertigen

Dietl has this right a lot of the time, but it does have

Testamentsvollstreckung: execution of a will

Dangerous! Romain knows better: executorship (but probably one can manage without the abstract noun). Take that, you people who sneer at Romain! (and thanks to Donna)

Execution has three meanings in English:

To quote the Oxford Dictionary of Law, now in its 6th edition (2006)

bq. Execution 1) The process of carrying out a sentence of death imposed by a court …
2) The enforcement of the rights of a judgment creditor …
3) The completion of the formalities necessary for a written document to become legally valid. In the case of a deed, for example, this comprises the signing and delivery of the document. [sealing is no longer required for a deed]

bq. Execution of a will: Under section 9 of the Wills Act 1837, the will must be signed at the end by the testator or by someone authorized by him, and the signature must be made or acknowledged by the testator in the present of at least two witnesses, present at the same time, who must themselves sign the will or acknowledge their signatures in the testator’s presence.

It’s confusing because the executor is the Testamentsvollstrecker (such an entity is essential in the common law, but just the icing on the cake in German law), not to be confused with the executioner (Henker, Scharfrichter).