"We made an offer for the goodwill and assets of the business and we were accepted,
said Rapier chairman Nick de Bois. "We have also managed to keep on most of the staff and have merged the design teams."
Blue Square's offices in east London have been retained for 12 months.
The agency's head of client services Christopher Curtis and head of creative design Laurent Nurisso remain employed by Rapier on a short-term basis.
de Bois added: "Blue Square is well-respected and operates in industries outside Rapier's core sectors, such as museums and interiors. As a business we are too heavily dependent on certain sectors so this is a natural way to diversify."
The Blue Square brand and identity will not be changed but will become a division of Rapier.
"We want the brand to survive,
explained de Bois. "It's a very well-established brand and what we've done is brought financial stability and additional resources to the business."
The new company is already working with former suppliers to the now defunct business. One of the projects it is working on is to create a 750 sq m experience for the Chelsea Flower Show later this month.
Last year Rapier won a two-year, £13m contract for BAE Systems' global exhibition business, including the Asian Aerospace exhibition, which took place in Singapore in February (Marketing Event, Nov/Dec).