Start at the town hall registry
Ask the Ufficio di Stato Civile at the Comune, the registry desk that logged your birth, for a fresh copy. A clerk prints it, signs it and adds the seal. A photocopy, a scan or an old print from a drawer will not do, because the stamp only goes onto a signed original.
Tell that clerk which version the office abroad asked for. A short extract lists the bare facts, while a full copy repeats every line the register holds, including later notes about marriage or a change of name. Picking the wrong one is the most common reason a file bounces, and a second trip to the counter costs days.
Why the Prefettura, and not the court
A birth record is an administrative paper, so the Prefettura, the state's own office for the province, is the desk that stamps it. Court copies and deeds signed before a notary are judicial or notarial, and those go to the Procura della Repubblica instead. Learning how to apostille a Birth Certificate mostly means learning which of the two counters is yours.
The province is set by the signature, not by you. To apostille a Birth Certificate in Genoa the registrar who signed the copy has to sit in that province, whatever passport you hold and wherever you live now. If your birth was logged in another town, that town's province owns the stamp.
The fee and the wait
The apostille cost for a Birth Certificate in Genoa is a €16 tax stamp per document, and some documents are exempt. Records going to another country in the European Union can fall under such an exemption, so ask at the counter which case yours is before you buy. How long you wait turns on the queue and on whether that office takes files by post or face to face.
What the stamped record is for
Most people need it to claim citizenship through a parent, to marry abroad, to enrol a child at a school, or to add a family member to a work file. A registry abroad wants proof that an Italian registrar really signed the page in front of them. Italy has sat inside the Hague Apostille Convention since 1978, and every state that signed it takes the single stamp.
Language, and what to check twice
The stamp says nothing at all about language. If the reading office works in another one, add a translation, and ask whether they want it sworn before a Tribunale, an Italian court, so the words count as accurate. The legalisation of a Birth Certificate in Genoa for a country outside the Hague list runs through a longer chain of offices, so settle the route in writing first.
Read the names, the dates and the place against your passport before anything is sent. One letter out of place, a dropped accent, or a date written the other way round is enough for a foreign clerk to hand it back. The rules to apostille a Birth Certificate in Italy are the same everywhere, but each desk abroad sets its own bar.