Short answer. Whoever alleges it. Past three hundred days from the termination of the marriage, the law supplies no presumption either way: legitimacy or illegitimacy must be proved by the party asserting it. If you claim the child is your late husband's, the proof is yours to bring.
What the law says
The legitimacy or illegitimacy of a child born after three hundred days following the termination of the marriage shall be proved by whoever alleges such legitimacy or illegitimacy.
Family Code, Article 169 — Children Born Beyond 300 Days After Termination. Read the full provision →
The presumption runs out at three hundred days
Within the marriage, and for three hundred days after it ends, the law presumes a child to be the husband's, and anyone who says otherwise must overturn the presumption. That period is a rough biological outer limit, and beyond it the law simply stops assuming. Eleven months is past the line. The consequence is not that your child is presumed illegitimate — the article is carefully neutral — but that no presumption operates at all, so the question is decided on evidence like any other question of fact.
The burden goes with the assertion
The article allocates the burden by reference to what is being claimed: it is proved by whoever alleges such legitimacy or illegitimacy. In practice this cuts both ways. A mother asserting that her late husband is the father must prove it; a relative of the deceased asserting the opposite, in order to exclude the child from the estate, must prove that. Nobody wins by default. Which side actually carries the burden in a given case therefore depends on who is asserting what, and often on who filed first.
What proof looks like
Since there is no presumption to lean on, the evidence must be capable of standing alone. The date of the husband's death against the date of birth is the starting point, and the medical record of the pregnancy — when it began, the estimated date of conception, whether the birth was premature or late — is usually the most valuable document in the file. Beyond that: proof that the spouses were living together during the relevant window, the birth certificate and who furnished its entries, and, where the question is genuinely contested, scientific evidence of parentage.
Why it will matter, and when
This question rarely stays theoretical, because the child's status decides participation in the estate and the size of the share. That is also when it tends to be raised — years later, by people with an interest in the answer, when the pregnancy records may be gone. So the useful thing to do now is preserve them: the prenatal records, the hospital file, the death certificate, the marriage certificate and the birth certificate, kept together. Then take the timeline to a lawyer while everything can still be located.