There are a number of non-chemical control options.
Dig, pull or hoe out plants before seed is set. Seedlings and plants that have not set seed can be added to the compost heap where the toxins will naturally break down. However, plants that have set seed should be consigned to the green waste collection, buried deeply (60cm/2ft or more) or burnt in order that the seed does not disperse in the garden or persist in the compost heap. Always wear gloves or thoroughly wash hands after handling this plant.
Use a mulch of organic matter, at least 5cm (2in) thick or opaque sheeting, such as woven polypropylene, to smother this weed.
Contact herbicides such as acetic acid (Weedol Fast Acting Gun), fatty acids (SBM Solabiol Super Fast Weedkiller) and pelargonic acid (Doff 24/7 Fast Acting Weedkiller) are good at knocking back young annual weeds such as thorn apple. Systemic herbicides such as glyphosate (e.g. Roundup Fast Action, SBM Job done General Purpose Weedkiller or Doff Glyphosate Weedkiller) will also be effective and are the material of choice where mature plants are to be eliminated.
These herbicides are non-selective and so care should be taken while spraying near other plants. Plants to be avoided can be covered with a bucket/flowerpot or screened with plastic when spraying. The gel formulation of glyphosate (Roundup Gel) is particularly useful for treating weeds among garden plants.
Inclusion of a weedkiller product does not indicate a recommendation or endorsement by the RHS. It is a list of products currently available to the home gardener.
Weedkillers for gardeners (Adobe Acrobat pdf document outlining weedkillers available to gardeners; see sections 3 and 4)
Chemicals: using a sprayer
Chemicals: using safely and effectively
Chemicals: using spot and broadscale weedkillers
Weeds: non-chemical control