No, mining will not come to an end at that point. The article is incorrect.
Mining will continue once the block rewards are no longer available, as transaction fees will continue to be offered. As the block rewards tend towards zero, the total value of transaction fees in each block will start to exceed the block reward. This will happen long before the block reward reaches zero if Bitcoin continues to be grow at the current rate.
Mining is required to provide the confirmation that each transaction is valid. If mining was to end, it would be because nobody was submitting transactions any more. This isn't anticipated any time soon!
This chart shows the value of transaction fees per day, in BTC. Currently, about 50BTC per day is paid in fees, which works out to be about 0.34BTC per block (50 divided by 144, which is the number of blocks per day).
If Bitcoin remains at it's currently level of popularity, transaction fees will be larger than the block reward by the year 2040. By the year 2064, more than 99% of miner's rewards would come from transaction fees. After 2140, there will be no more block rewards given.