Will I be able to register and trademark this business name?

There are a lot of other facts that you would have to know to answer your question. Trademark rights are acquired through use in connection with the offering and sale of goods/services so that consumers can identify and distinguish the source of those goods/services from other providers of similar goods/services. A user of a trademark has common law rights by virtue of using the trademark, and those common law rights are limited to the geographic area in which the mark is used. It can be very difficult to determine where that geographic area lies. Federal registration gives a trademark user presumptively nationwide rights. Under your facts, it is not clear whether the other company has a Federal registration or where the other company's common law rights exist. It is also not clear whether another person has registered or is using a trademark that is similar to your trademark. A business name does not necessarily establish trademark rights, so it is also not clear whether you are referring to a "trade name" or a trademark. One final point, goods/services do not have to be identical to be infringing, they only have to be related or in the natural zone of expansion of the senior (prior) user of that trademark. You should talk to an experienced trademark lawyer to decide if it is worth investing in this trademark if there is a possibility that you would have to stop using it in the future.