The high-level industry appointment sees the TV trade body step up ahead of an expansion programme that began with its budget being increased to £3m and will be followed with the hiring of a full staff.
Alps, who will join the organisation in July, will work closely with Thinkbox members, which include Channel 4, Five, ITV and Sky, to push the case for commercial TV.
Other candidates who were believed to have been considered included BT's head of media, Steve Huddleston.
Alps's appointment comes after a two-month search and will see her pick up a salary in the region of between £200,000 and £250,000.
The hunt for a chief executive kicked off when ITV parted ways with Justin Sampson, its director of customer relationship marketing, making his dual role as the Thinkbox chairman untenable.
Alps said that Thinkbox for her was in many ways her dream job and the only one she would have left PHD for. However, she realises the major challenge that she faces.
"I don't underestimate the challenge, but I will be on a mission to help advertisers and agencies get more out of the amazing medium that is television. This has never been stronger and technology is helping people access more television in new ways that fit their lives.
"Brands can be part of this in so many compelling and effective ways. I want to help all advertisers cut through the hysterical nonsense that is frequently peddled by competitors to reap the benefits of TV for their business," she said.
Her exit from PHD brings to an end a long career with the media planning and buying agency.
It takes her back to where she started her career, in TV sales, before she became ITV sales director in 1989.
Andy Barnes, Channel 4 head of Channel 4 ad sales and Thinkbox chairman, said: "To be able to attract someone of Tess's quality and pedigree to the position of CEO is quite simply stunning. Stunning that we have been able to attract her and stunning that she is able to take on the role.
"All of which is a stunning endorsement of what Thinkbox has established and achieved to date. She will play an integral role in developing Thinkbox's strategy and creating a marketing body that truly delivers on the needs of advertisers today."
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