In the last couple of months, Donald Trump has been boasting that "Black Unemployment has just been reported to be at the LOWEST RATE EVER RECORDED!" At the same time, a report from the Economic Policy Institute says that black unemployment is higher than it was fifty years ago. The figures for black unemployment, which I got via FRED, start in January 1972. The 6.8% in December 2017 is indeed the lowest since those records began (the previous low was 7.0% in April 2000).
The EPI numbers refer to unemployment for "black and other," and were 6.7% in 1968 and 6.4% in 1969 (they give a link to the table). Like the figures on unemployment among blacks, they are from the Bureau of Labor Statistics, so the question is how much difference including "other" makes. They report both "black and other" and "black" for 1972, and "black" is 0.4% higher (10.4% vs. 10%). If you estimate unemployment among blacks in 1968 and 1969 by adding 0.3 or 0.4, it was about the same as it is today.
Of course, white unemployment is low now and was low in the late 1960s, so if you are interested in racial differences, the thing to look at is the ratio of black unemployment rate to white unemployment rate.
The ratio bounces around from one month to the next, presumably mostly because of sampling error, so I also show a 23-month moving average in red (chosen because it seemed to give a reasonable balance between simplicity and detail). It seems like the ratio increased somewhat through the late 1980s and then fell, leaving it just slightly lower at the end than the beginning. Things are a bit more complicated than that, because the ratio tends to be higher when general unemployment is low, as it is now--if you adjust for that, there is more of a downward trend in the ratio. Still, the trend is not as large as I would have expected given the narrowing of racial differences in education.