American Express Co. and Shearson Loeb Rhoades Inc. announced yesterday they have agreed to a $900 million merger, in which Shearson, the nation's second-biggest brokerage house, will become a subsidiary of the giant financial services community.

Shearson, with about $470 million in capital and a nationwide network of 4,000 brokers, will become the second major Wall Street firm acquired by a company that is not a broker. Last month, Prudential Insurance Co. announced it planned to spend $385 million -- $32 a share -- to buy the Bache Group Inc., the eight-largest brokerage firm in the country. Monday, Prudential said it had acquired 70 percent of Bache.

Wall Street observers predict there will be more mergers between brokerage houses and financial service companies -- such as insurers -- as the distinctions between different financial products blur.

For example, American Express owns Fireman's Fund Insurance Company, whose products can be marketed through Shearson brokers. Similarly, Shearson, like most other major brokers, has been developing an account similar to Merrill Lynch & Co.'s cash management accounts. Merrill Lynch is the nation's largest brokerage firm.

Under the Merrill plan, customers with money on deposit in a cach management account can earn interest and write checks on funds that under normal brokerage practices would lie idle until the customer withdrew them or used them to purchase more stock. The Merrill customers also have a Visa credit card with their accounts.

Presumably, American Express travel cards could be used with whatever type of cash management account Shearson develops.

American Express Chairman James D. Robinson III said that American Express plans to exchange 1.3 shares of its stock for each of the 15.6 million outstanding shares of Shearson stock. Shareholders of both companies must approve the merger -- as well as the boards of directors of both and regulatory authorities.

Robinson said he anticipates the merger will take 60 days to complete.

Robinson and Shearson Chairman Sanford I. Weill first met to discuss possible joint ventures between the two firms last fall, Robinson said. Lower-level talks between the two firms continued until last month, when Prudential and Bach announced their merger agreement, Robinson said. That merger triggered a more serious discussion between American Express and Shearson, Robinson said.

At a senior level, executives explored whether the firms would mesh and whether their business philosophies were compatible. "They were," Robinson said.

"What is normally the hardest part was the easiest," Weill said. He said that the firms' philosophies meshed so much "trust" in one another through the months of discussions that deciding upon the "structure of the transaction" was simple.

Shearson will become a wholly owned, independent subsidiary of American Express. Weill will remain chairman of Shearson and will become a director of the parent company and chairman of American Express' executive committee. The companies said that Weill would join Robinson and three other American Express excutives in an expanded executive office of the chairman.

American Express, which last year had revenues of $5.5 billion, is best-known for its travelers check and entertainment cards. But the company also owns a large foreign bank in addition to the insurance company.

Shearson Loeb Rhoades, like most major brokerage firms, is a composite of many formerly prestigious Wall Street houses that, during the difficult years of the 1970s, merged into stronger brokerage companies.