Stocktrade is being transferred to the Dundee-headquartered firm for £14m in cash.
Its most recent annual revenue was £9m, with £1.3m in pre-tax profit.
Last September, Stocktrade had £4.3bn of assets under management. The buyer said it now stands at £4.6bn, bringing 48,000 new customers.
That should increase the scale of Alliance Trust Savings to £11.5bn in assets under management, with 105,000 customers, allowing for more efficient operations.
It will also increase the Dundonians range of services, including telephone and online share dealing services for pension providers, building societies, company share schemes and corporate Individual Savings Accounts.
Brewin Dolphin said the sale would only clear a net gain of £1m, after paying for separation costs.
The deal still requires clearance from regulators, and is expected to close during this year.
Alliance Trust Savings was part of the strategy last month criticised by Elliott Advisers, the hedge fund and activist investor in parent company Alliance Trust.
In criticising the Dundee company's management, it succeeded in getting two new directors appointed. They could have sought to block further expansion of the financial retail platform. But it appears, from this deal, that they have agreed to expenditure on growth.
Alliance Trust believes that the expansion could help its savings division into profit by 2016, apart from the cost of integration.
Patrick Mill, managing director of Alliance Trust Savings (ATS), said: "Stocktrade is a great strategic fit for ATS, giving us access to new customers, an increase in assets and the opportunity to build a strategic partnership capability.
"It also gives us the ability to align a traditional stockbroking business with our innovative, scalable and cost effective approach by offering customers online access alongside our in-house customer service team."
Alliance Trust Savings has announced plans to buy an Edinburgh-based share trading company from Brewin Dolphin.