The takeover is a bid to bolster Lloyds' consumer finance offerings and will complete in the first half of 2017.
It will add around £650m a year to group revenues, Lloyds said in a statement, and is also expected to have "significant opportunity" for cost synergies, expected at a £100m run rate within two years.
In the first half of 2016 MBNA's post-tax profits came in at £123m. The £1.9bn purchase price includes £800,000 of acquired equity and £240m for future payment protection insurance (PPI) claims.
Bank of America had originally refused to share the future costs of compensation in relation to mis-selling PPI. At one point, the PPI exposure was reported to be putting Lloyds off from buying MBNA.
But the US lender has now agreed to indemnify the new owner of the business if costs rise above a fixed cap.
Lloyds' group chief executive Antonio Horta-Osorio said:
The acquisition, funded through strong internal capital generation, increases our participation in the expanding UK credit card market with a multi-brand strategy and advances our strategic aim to deliver sustainable growth as a UK focused retail and commercial bank.
The MBNA brand and portfolio are a good fit with our existing card business and we will focus on providing its customers with excellent service and value. Our low cost to income ratio and proven integration capabilities will deliver significant synergies and value to our shareholders.