Having made a dent in the pirating of their products at home, American makers of software for personal computers are turning their attention abroad. They are mobilizing lawyers, big budgets and the threat of unwanted publicity to attack a practice that they say costs them billions of dollars in lost exports each year.

Earlier this month the big French financial institution Banque Paribas became the latest offender to give in. It settled a suit brought by U.S. companies accusing it of making wide- spread use of bootlegged programs in PCs in its offices and paid for illicit copies of the programs.

With victories like these to wave around, software industry officials contend that the battle is being won, if slowly. "We pick a country where we think we can have leverage," said Microsoft Corp. chairman William Gates, citing Italy, where several lawsuits were filed. "Our growth in sales is due to the reduction of piracy. That's the biggest reason ... my Italian subsidiary has grown."

Still, the companies are hardly happy. Richard Neff, deputy general counsel of the software firm Ashton-Tate Corp., tells of finding a wide assortment of illicit U.S. programs while visiting computer shops in Thailand and South Korea recently. A copy of dBASE, his company's flagship product, was available in one Thai shop for less than a tenth of its U.S. price.

Countries generally cover computer software under the same copyright laws that outlaw the unauthorized duplication of books. Though interpretations vary, the computer industry and many copyright experts say that computer users cannot copy and distribute the software they buy. If a company has 50 PCs that need to run dBASE simultaneously, for example, it must buy 50 programs at full price.

Of major PC programs, said Neff, "there are lots more pirated copies than legitimate copies in use worldwide."

The allure in international markets of many U.S. manufactured goods has eroded in recent times. But if its hardware is troubled, the country's software -- which in the broad sense includes recorded music, books and movies as well as computer programs -- remains a blockbuster, setting the pace for the world. But it is also being copied en masse.

Estimating the cost of piracy is a tricky business, because people who buy the cut-rateversions might simply do without if only the full-price types were available. But there is evidence that the problem is enormous.

Figures compiled by the Business Software Alliance (BSA), which is bankrolled by six big U.S. firms that make PC software, show that for every three computers sold in Italy, only one legitimate program is sold, suggesting the other two use something from under the table. The group estimates that in sum U.S. software companies lose about $5 billion in sales per year in Europe alone.

U.S. government trade negotiators have made protection of these and other forms of "intellectual property" a prime goal in one-on-one talks in foreign capitals. Through the Geneva-based organization General Agreement on Tariffs and Trade, they have also tried to get the general rules of world commerce changed to put greater stress on it.

U.S. PC software firms, which had sales of more than $12 billion abroad in 1989, have not sat waiting in the meantime. Unlike software for mainframe computers, which is generally customized to specific machines, their product for small machines is readily transferable. So they have sent people abroad to work within the current laws. Most visible is the Washington-based BSA.

Many computer users, abroad as well as in the United States, see no harm in making copies. They may not know it is illegal in most countries under copyright law. Or they may feel little sympathy for what they see as the large, wealthy and in many cases foreign company that originally produced it. The task itself is simple, since every PC is also a copying device, capable of cloning in minutes programs that may have taken years and millions of dollars to develop.

In some developing countries, officials and business leaders argue privately that local companies need the cost advantage of cheap software if they are to compete with larger, more experienced foreign companies. Piracy is seen as a legitimate way of bringing a bit of the know-how of the industrialized world inside their borders. Thus anti-piracy laws on local books may not be enforced.

In some cases, fear of piracy can cool U.S. companies' enthusiasm for selling overseas. "Everyone has told us that if you sell foreign, things will get copied," said Lyle Warrington, president of Minnesota-based Red Wing Business Systems, which is cautiously starting to sell in Mexico its accounting software for farms.

Some of BSA's main targets are mass producers of pirated software. In Hong Kong a year ago, for instance, a raid netted close to 110,000 pirated software manuals, as well as thousands of floppy disks.

In Europe, the focus is more on offices that buy one legitimate copy of a program and then replicate it to be used on many PCs inside the organization. Earlier this month BSA and other groups sued Marconi Instruments Ltd. for alleged software infringement. In France BSA conducted court-authorized inspections of two companies, Rhone-Poulenc Films and France Distribution Systems.

BSA is also lobbying to strengthen local laws and trying to raise awareness. BSA distributes pamphlets on software protection in Italian, French, Chinese and Korean to company computer supervisors. It has lobbied the European Community for a Europe-wide approach to software protection, and it is proposing ideas for Eastern Europe, where PCs are proliferating.

Software makers have waged a similar drive against piracy in the United States. Ken Wasch, executive director of the Software Publishers Association (SPA), says his association almost never hears complaints any more of computer dealers giving out illicit copies of programs to help sell machines.

The SPA has passed out free of charge more than 15,000 copies of a program it developed. Slipped into different computers successively, it builds a running total for managers of how many copies of various programs it has found.

The association also runs a hotline. Mostly on the basis of tips called in, the BSA has conducted more than 50 "audits" of suspected offender companies. The companies agree, generally under threat of lawsuit, to have their PCs inspected and to pay penalties for illicit copies found.

Several years ago, said Wasch, "it was easy for a company to say, 'sure it's illegal but who's coming after me?' Now the answer here and throughout Europe is, 'we will.' "