Short answer. An administrator. Where an heir is instituted under a suspensive condition or term, the estate is placed under administration until the condition is fulfilled, until it becomes certain it cannot be, or until the term arrives. The same happens if the heir fails to give a required security. No one takes the property outright in the meantime.
What the law says
If the heir be instituted under a suspensive condition or term, the estate shall be placed under administration until the condition is fulfilled, or until it becomes certain that it cannot be fulfilled, or until the arrival of the term.
Civil Code, Article 880 — Administration Pending a Condition. Read the full provision →
The estate goes under administration
Article 880 covers the gap that opens when an inheritance is not yet due: if the heir be instituted under a suspensive condition or term, the estate shall be placed under administration until the condition is fulfilled, or until it becomes certain that it cannot be fulfilled, or until the arrival of the term. A suspensive condition means the heir inherits only if some uncertain future event happens; a term means he inherits at a set future time. Until that point is reached, the conditionally instituted heir has no present right to hold the property, so it is entrusted to an administrator rather than delivered.
Why an interim custodian is needed
The property cannot simply sit unmanaged, nor can it be handed to an heir whose entitlement has not yet arisen. Someone must preserve it, collect its fruits, pay what must be paid, and keep it intact so that whoever is ultimately entitled receives it in good order. Administration supplies that neutral custodian for the waiting period. It protects everyone with a possible interest, the conditional heir, and those who would take if the condition fails, without prejudging which of them ends up owning the property.
When the administration ends
The article marks three end points. Administration continues until the condition is fulfilled, at which the heir's right becomes absolute and the property can pass to him; or until it becomes certain the condition cannot be fulfilled, at which the conditional gift fails and the property goes to whoever is then entitled; or, in the case of a term, until the term arrives. Each is a moment at which the uncertainty resolves, and the interim custody gives way to delivery to the person the resolution favours.
Administration also follows a missing security
The final sentence ties this to the security rules: the estate is likewise placed under administration if the heir does not give the security required in the preceding article. So an heir who is entitled to take conditionally burdened property early, but who fails to post the bond the law asks for, does not simply keep the property unsecured; it is administered instead until matters are put right. If you are dealing with a pending conditional inheritance, expect the assets to be held by an administrator, and address any required security through the settlement proceeding.