Bill and Melinda Gates gave the last $15 million to complete a computer-science building at the University of Washington, which just happens to be named after them — although UW officials say that’s purely a coincidence.

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Even in Seattle these days, it takes considerable work to raise $70 million in private donations for a new classroom and lab building on the University of Washington campus.

And so, despite a two-year campaign and multimillion-dollar gifts from Microsoft, Amazon, Zillow, Google and well-heeled private donors, the Allen School of Computer Science & Engineering was still short — by $15 million — of the money it had pledged to raise for its new building by the end of this year.

Then Bill and Melinda Gates offered to make up the difference, said computer-science professor Ed Lazowska, who has helped lead the private-funding effort.

The irony: A small group of well-heeled funders, calling themselves “Friends of Bill and Melinda,” had already talked about naming the building after the Gateses.

Purely coincidental, Lazowska insists.

Holding out naming rights to the Gateses in exchange for their names on the building “would have been clever, but we weren’t devious enough to think of it,” he said via email. When the UW Regents voted Oct. 12 to name the building after the couple — officially designating it the Bill & Melinda Gates Center — the university said they had not donated any money to the building.

On Wednesday, the day the final steel beam is to go into place, UW officials announced that the Gateses had, in fact, donated to the building, giving the UW one of the biggest single donations for the project.

The donation happened around the same time the building was named and was “entirely independent” of the naming decision, Lazowska said. “We wanted to celebrate the naming and the Friends of Bill and Melinda, without confusing this with the entirely separate gift from the Gateses.”

In all, more than 350 people donated to the project, including many contributions from alumni.

In a statement, Microsoft President Brad Smith, who chaired the fundraising drive and contributed money of his own, called the new building “vital to Washington.” It will allow the UW to double the number of students who can major in computer science, one of the most sought-after majors on campus.

The UW plans to hold a “topping out” ceremony Wednesday afternoon to celebrate the placement of the final steel beam at the top of the structure. It marks the halfway point in construction; contractors will now begin working on enclosing and building out the interior. The building is scheduled to be completed in early 2019.

The building will include classrooms, a robotics lab, a wet lab for research involving computing and biology, an undergraduate commons and office and collaboration spaces. It is across the street from another building named after a Microsoft founder — the Paul G. Allen Center for Computer Science & Engineering.

Of the $110 million cost, $17.5 million came from the state’s 2015-17 capital budget. An additional $15 million came from the UW’s own building account, which is funded by tuition dollars and revenue generated by the land the UW owns in downtown Seattle, known as the Metro Tract. And the UW provided an additional $9 million from its own budget.